# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

### 2.1.5 Lax Monoidal Functors

We now introduce a unital version of Definition 2.1.4.3. To motivate the discussion, we begin with a special case.

Definition 2.1.5.1. Let $\operatorname{\mathcal{C}}$ be a monoidal category with unit object $\mathbf{1}$, and let $A$ be a nonunital algebra object of $\operatorname{\mathcal{C}}$ (Example 2.1.4.11) with multiplication $m: A \otimes A \rightarrow A$. We say that a morphism $\epsilon : \mathbf{1} \rightarrow A$ is a left unit for $A$ if the composite map

$A \xrightarrow { \lambda _{A}^{-1} } \mathbf{1} \otimes A \xrightarrow { \epsilon \otimes \operatorname{id}_ A} A \otimes A \xrightarrow {m} A$

is the identity map from $A$ to itself; here $\lambda _{A}: \mathbf{1} \otimes A \xrightarrow {\sim } A$ denotes the left unit constraint of Construction 2.1.2.17. We say that $\epsilon$ is a right unit of $A$ if the composite map

$A \xrightarrow { \rho _{A}^{-1} } A \otimes \mathbf{1} \xrightarrow { \operatorname{id}_{A} \otimes \epsilon } A \otimes A \xrightarrow {m} A$

is equal to the identity. We say that $\epsilon$ is a unit of $A$ if it is both a left and a right unit of $A$.

By virtue of Example 2.1.4.11, we can view the theory of nonunital algebras as a special case of the theory of nonunital lax monoidal functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where we take $\operatorname{\mathcal{C}}$ to be the trivial monoid $\{ e \}$ (regarded as a category having only identity morphisms). Definition 2.1.5.1 has an analogue for nonunital lax monoidal functors in general.

Definition 2.1.5.2. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories with unit objects $\mathbf{1}_{\operatorname{\mathcal{C}}}$ and $\mathbf{1}_{\operatorname{\mathcal{D}}}$, respectively. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor with tensor constraints $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$. Let $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F(\mathbf{1}_{\operatorname{\mathcal{C}}})$ be a morphism in $\operatorname{\mathcal{D}}$. We say that $\epsilon$ is a left unit for $F$ if, for every object $X \in \operatorname{\mathcal{C}}$, the left unit constraint $\lambda _{F(X)}: \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes F(X) \xrightarrow {\sim } F(X)$ in the category $\operatorname{\mathcal{D}}$ is equal to the composition

$\mathbf{1}_{\operatorname{\mathcal{D}}} \otimes F(X) \xrightarrow { \epsilon \otimes \operatorname{id}_{F(X)}} F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes F(X) \xrightarrow { \mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, X} } F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes X) \xrightarrow { F( \lambda _ X) } F(X),$

where $\lambda _ X: \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes X \xrightarrow {\sim } X$ is the left unit constraint in the monoidal category $\operatorname{\mathcal{C}}$. We say that $\epsilon$ is a right unit for $F$ if, for every object $X \in \operatorname{\mathcal{C}}$, the right unit constraint $\rho _{F(X)}: F(X) \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \xrightarrow {\sim } F(X)$ is equal to the composition

$F(X) \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \xrightarrow { \operatorname{id}_{F(X)} \otimes \epsilon } F(X) \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}}) \xrightarrow { \mu _{X, \mathbf{1}_{\operatorname{\mathcal{C}}}} } F( X \otimes \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { F( \rho _{X} ) } F(X).$

We say that $\epsilon$ is a unit for $F$ if it is both a left and a right unit for $F$.

Example 2.1.5.3. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $A$ be a nonunital algebra object of $\operatorname{\mathcal{C}}$, which we identify with a nonunital lax monoidal functor $F: \{ e\} \rightarrow \operatorname{\mathcal{C}}$ as in Example 2.1.4.11. Then a map $\epsilon : \mathbf{1} \rightarrow A = F(e)$ is a unit (left unit, right unit) for $A$ (in the sense of Definition 2.1.5.1) if and only if it is a unit (left unit, right unit) for $F$ (in the sense of Definition 2.1.5.2).

We now show that if a nonunital lax monoidal functor $F$ admits a unit $\epsilon$, then $\epsilon$ is uniquely determined. This is a consequence of the following:

Proposition 2.1.5.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories with unit objects $\mathbf{1}_{\operatorname{\mathcal{C}}}$ and $\mathbf{1}_{\operatorname{\mathcal{D}}}$, respectively, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor. Suppose that $F$ admits a left unit $\epsilon _{L}: \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F(\mathbf{1}_{\operatorname{\mathcal{C}}})$ and a right unit $\epsilon _{R}: \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$. Then $\epsilon _{L} = \epsilon _{R}$.

Proof. We first observe that there is a commutative diagram

$\xymatrix@C =50pt{ \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{ \operatorname{id}\otimes \epsilon _{R} } \ar [dd]^{ \lambda _{ \mathbf{1}_{\operatorname{\mathcal{D}}} } }& \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \ar [ddr]_-{ \lambda _{ F(\mathbf{1}_{\operatorname{\mathcal{C}}}) } } \ar [r]^-{ \epsilon _{L} \otimes \operatorname{id}} & F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \ar [d]^{\mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}} , \mathbf{1}_{\operatorname{\mathcal{C}}} }} \\ & & F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes \mathbf{1}_{\operatorname{\mathcal{C}}} ) \ar [d]^{ F( \lambda _{ \mathbf{1}_{\operatorname{\mathcal{C}}} }) } \\ \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [rr]^{ \epsilon _{R} } & & F( \mathbf{1}_{\operatorname{\mathcal{C}}} ); }$

the left square commutes by the naturality of the left unit constraints for $\operatorname{\mathcal{C}}$ (Remark 2.1.2.18), and the right square commutes by virtue of our assumption that $\epsilon _{L}$ is a left unit for $\operatorname{\mathcal{C}}$. Using Corollary 2.1.2.21, we see that the unit constraints

$\upsilon _{\operatorname{\mathcal{C}}}: \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes \mathbf{1}_{\operatorname{\mathcal{C}}} \xrightarrow {\sim } \mathbf{1}_{\operatorname{\mathcal{C}}} \quad \quad \upsilon _{\operatorname{\mathcal{D}}}: \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \xrightarrow {\sim } \mathbf{1}_{\operatorname{\mathcal{D}}}$

are equal to the left unit constraints $\lambda _{ \mathbf{1}_{\operatorname{\mathcal{C}}} }$ and $\lambda _{ \mathbf{1}_{\operatorname{\mathcal{D}}} }$, respectively. It follows that the composition $\epsilon _{R} \circ \upsilon _{\operatorname{\mathcal{D}}}$ coincides with the composition

$\mathbf{1}_{\operatorname{\mathcal{D}}} \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \xrightarrow { \epsilon _{L} \otimes \epsilon _{R} } F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { \mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, \mathbf{1}_{\operatorname{\mathcal{C}}} } } F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { F( \upsilon _{\operatorname{\mathcal{C}}} )} F( \mathbf{1}_{\operatorname{\mathcal{C}}} ).$

A similar argument shows that this composition coincides with $\epsilon _{L} \circ \upsilon _{\operatorname{\mathcal{D}}}$. Since $\upsilon _{\operatorname{\mathcal{D}}}$ is an isomorphism, it follows that $\epsilon _{R} = \epsilon _{L}$. $\square$

Corollary 2.1.5.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor. Then $F$ admits a unit $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F(\mathbf{1}_{\operatorname{\mathcal{C}}} )$ if and only if it has both a left unit and a right unit. In this case, the unit $\epsilon$ is unique.

Proposition 2.1.5.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories with unit objects $\mathbf{1}_{\operatorname{\mathcal{C}}}$ and $\mathbf{1}_{\operatorname{\mathcal{D}}}$, respectively. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor equipped with a nonunital lax monoidal structure, which we will identify with the corresponding nonunital monoidal structure on the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ (see Proposition 2.1.4.21). Let $\epsilon : \mathbf{1}_{\operatorname{\mathcal{C}}} \rightarrow G( \mathbf{1}_{\operatorname{\mathcal{D}}} )$ be a morphism in $\operatorname{\mathcal{C}}$, and regard the triple $\mathbf{1} = ( \mathbf{1}_{\operatorname{\mathcal{C}}}, \mathbf{1}_{\operatorname{\mathcal{D}}}, \epsilon )$ as an object of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. Then:

$(1)$

The morphism $\epsilon$ is a left unit for $G$ if and only if, for every object $(C, D, \eta )$ of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, the left unit constraints $\lambda _{C}: \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes C \simeq C$ and $\lambda _{D}: \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes D \simeq D$ determine an isomorphism $(\lambda _ C, \lambda _ D): \mathbf{1} \otimes (C,D, \eta ) \simeq (C,D,\eta )$ in the category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$.

$(2)$

The morphism $\epsilon$ is a right unit for $G$ if and only if, for every object $(C, D, \eta )$ of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, the right unit constraints $\rho _{C}: C \otimes \mathbf{1}_{\operatorname{\mathcal{C}}} \simeq C$ and $\rho _{D}: D \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \simeq D$ determine an isomorphism $(\rho _ C, \rho _ D): (C,D, \eta ) \otimes \mathbf{1} \simeq (C,D,\eta )$ in the category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Fix an object $(C,D, \eta )$ of the oriented fiber product $\operatorname{\mathcal{C}}\otimes _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. Unwinding the definitions, we see that the pair $(\lambda _ C, \lambda _ D)$ determines a morphism from $\mathbf{1} \otimes (C,D,\eta )$ to $(C,D, \eta )$ in $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ if and only if the outer rectangle of the diagram

$\xymatrix@R =50pt@C=50pt{ \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes C \ar [r]^-{ \lambda _{C} } \ar [d]^{ \operatorname{id}\otimes \eta } & C \ar [d]^{\eta } \\ \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes G(D) \ar [r]^-{ \lambda _{G(D)}} \ar [d]^{ \epsilon \otimes \operatorname{id}} \ar [r] & G(D) \ar [dd]^{\operatorname{id}} \\ G( \mathbf{1}_{\operatorname{\mathcal{D}}} ) \otimes G(D) \ar [d]^{\mu } & \\ G( \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes D) \ar [r]^-{ G( \lambda _ D ) } & G(D) }$

is commutative. Here the upper square commutes by the functoriality of the left unit constraints in $\operatorname{\mathcal{C}}$ (Remark 2.1.2.18), and the commutativity of the lower rectangle follows from the assumption that $\epsilon$ is a left unit. This proves the “only if” direction of $(1)$. The converse follows by specializing to the case where $C = G(D)$ and $\eta$ is the identity map. $\square$

Corollary 2.1.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories with units $( \mathbf{1}_{\operatorname{\mathcal{C}}}, \upsilon _{\operatorname{\mathcal{C}}})$ and $(\mathbf{1}_{\operatorname{\mathcal{D}}}, \upsilon _{\operatorname{\mathcal{D}}})$, respectively. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a nonunital lax monoidal functor. Let $\epsilon : \mathbf{1}_{\operatorname{\mathcal{C}}} \rightarrow G( \mathbf{1}_{\operatorname{\mathcal{D}}} )$ be a morphism in $\operatorname{\mathcal{C}}$ and regard the triple $\mathbf{1} = ( \mathbf{1}_{\operatorname{\mathcal{C}}}, \mathbf{1}_{\operatorname{\mathcal{D}}}, \epsilon )$ as an object of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. The following conditions are equivalent:

$(1)$

The morphism $\epsilon$ is a unit for $G$ (in the sense of Definition 2.1.5.2).

$(2)$

The pair $\upsilon = (\upsilon _{\operatorname{\mathcal{C}}}, \upsilon _{\operatorname{\mathcal{D}}})$ is a morphism from $\mathbf{1} \otimes \mathbf{1}$ to $\mathbf{1}$ in the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, and the pair $(\mathbf{1}, \upsilon )$ is a unit with respect to the tensor product $\otimes _{\mu }$ of Proposition 2.1.4.21.

Proof. Assume first that $(1)$ is satisfied. Then Proposition 2.1.5.6 implies that the functors

$\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\quad \quad X \mapsto \mathbf{1} \otimes X, X \mapsto X \otimes \mathbf{1}$

are naturally isomorphic to the identity, and are therefore fully faithful. To complete the proof of $(2)$, it will suffice to show that the pair $( \upsilon _{\operatorname{\mathcal{C}}}, \upsilon _{\operatorname{\mathcal{D}}} )$ is a morphism from $\mathbf{1} \otimes \mathbf{1}$ to $\mathbf{1}$ in $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. This also follows from Proposition 2.1.5.6, by virtue of the identities $\upsilon _{\operatorname{\mathcal{C}}} = \lambda _{ \mathbf{1}_{\operatorname{\mathcal{C}}} }$ and $\upsilon _{\operatorname{\mathcal{D}}} = \lambda _{ \mathbf{1}_{\operatorname{\mathcal{D}}} }$ (Corollary 2.1.2.21).

Now suppose that $(2)$ is satisfied, so that we can regard $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ as a monoidal category with unit $( \mathbf{1}, \upsilon )$. It follows that the forgetful functor $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ carries the left and right unit constraints of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ to the left and right unit constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. Applying Proposition 2.1.5.6, we conclude that $\epsilon$ is both a left and right unit for the nonunital lax monoidal functor $G$. $\square$

Definition 2.1.5.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. A lax monoidal structure on $F$ is a nonunital lax monoidal structure $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ (Definition 2.1.4.3) for which there exists a unit $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$.

A lax monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a pair $(F, \mu )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is functor and $\mu$ is a lax monoidal structure on $F$. In this case, we will refer to the morphism $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$ as the unit of $F$.

Remark 2.1.5.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor. The condition that $F$ is a lax monoidal functor depends only on the underlying nonunital monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, and not on the particular choice of units $(\mathbf{1}_{\operatorname{\mathcal{C}}}, \upsilon _{\operatorname{\mathcal{C}}})$ and $(\mathbf{1}_{\operatorname{\mathcal{D}}}, \upsilon _{\operatorname{\mathcal{D}}})$ for $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively (see Remark 2.1.2.11).

Combining Proposition 2.1.4.21 with Corollary 2.1.5.7, we obtain the following:

Corollary 2.1.5.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor, let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ be the oriented fiber product of Notation 2.1.4.19, and let $U: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ denote the forgetful functor $(C,D, \eta ) \mapsto (C,D)$. Then the construction $\mu \mapsto \otimes _{\mu }$ of Proposition 2.1.4.21 restricts to a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Lax monoidal structures on G} \} \ar [d] \\ {\begin{Bmatrix} \textnormal{Monoidal structures on \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}} \\ \textnormal{with U strict monoidal} \end{Bmatrix}} }$

(see Example 2.1.6.5).

Variant 2.1.5.11. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. A colax monoidal structure on $F$ is a lax monoidal structure on the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$: that is, a collection of maps $\mu = \{ \mu _{X,Y}: F(X \otimes Y) \rightarrow F(X) \otimes F(Y) \} _{X,Y \in \operatorname{\mathcal{C}}}$ satisfying the requirements of Variant 2.1.4.16, together with the additional condition that there exists a counit $\epsilon : F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \rightarrow \mathbf{1}_{\operatorname{\mathcal{D}}}$ having the property that, for every object $X \in \operatorname{\mathcal{C}}$, the left and right unit constraints of $F(X)$ the inverses of the composite maps

$F(X) \xrightarrow { F(\lambda _ X)} F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes X) \xrightarrow { \mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, X} } F( \mathbf{1}_{\operatorname{\mathcal{C}}}) \otimes F(X) \xrightarrow { \epsilon \otimes \operatorname{id}} \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes F(X)$
$F(X) \xrightarrow { F(\rho _ X)} F( X \otimes \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { \mu _{ X, \mathbf{1}_{\operatorname{\mathcal{C}}} } } F(X) \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { \operatorname{id}\otimes \epsilon } F(X) \otimes \mathbf{1}_{\operatorname{\mathcal{C}}}.$

Remark 2.1.5.12 (Adjoint Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and suppose we are given a pair of adjoint functors $\xymatrix@1{\operatorname{\mathcal{C}} \ar@ <.4ex>[r]^-{F} & \operatorname{\mathcal{D}} \ar@ <.4ex>[l]^-{G}}$, given by an isomorphism of oriented fiber products $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ (see Notation 2.1.4.19). Applying Corollary 2.1.5.10 (and the dual characterization of colax monoidal functors), we see that the following are equivalent:

• The datum of a lax monoidal structure on the functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

• The datum of a colax monoidal structure on the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

• The datum of a monoidal structure on the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ which is compatible with the monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.

The compatibility conditions appearing in Definition 2.1.5.2 can be formulated more directly in terms of the unit constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ (without referring the left and right unit constraints of Construction 2.1.2.17).

Proposition 2.1.5.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories with unit objects $\mathbf{1}_{\operatorname{\mathcal{C}}}$ and $\mathbf{1}_{\operatorname{\mathcal{D}}}$, respectively, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor, and let $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$ be a morphism in $\operatorname{\mathcal{C}}$. Then $\epsilon$ is a left unit for $F$ if and only if it satisfies the following pair of conditions:

$(1)$

The diagram

$\xymatrix@R =50pt@C=50pt{ \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{ \epsilon \otimes \epsilon } \ar [dd]^{ \upsilon _{\operatorname{\mathcal{D}}} } & F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \ar [d]^{ \mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, \mathbf{1}_{\operatorname{\mathcal{C}}} }} \\ & F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes \mathbf{1}_{\operatorname{\mathcal{C}}}) \ar [d]^{ F( \upsilon _{\operatorname{\mathcal{C}}}) } \\ \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{ \epsilon } & F( \mathbf{1}_{\operatorname{\mathcal{C}}}) }$

commutes (in the category $\operatorname{\mathcal{D}}$). Here $\upsilon _{\operatorname{\mathcal{C}}}$ and $\upsilon _{\operatorname{\mathcal{D}}}$ denote the unit constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively.

$(2)$

For every object $X \in \operatorname{\mathcal{C}}$, the composite map

$\mathbf{1}_{\operatorname{\mathcal{D}}} \otimes F(X) \xrightarrow {\epsilon \otimes \operatorname{id}_{F(X)} } F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes F(X) \xrightarrow {\mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, X} } F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes X)$

is a monomorphism in the category $\operatorname{\mathcal{C}}$.

Moreover, if these conditions are satisfied, then the map

$\mathbf{1}_{\operatorname{\mathcal{D}}} \otimes F(X) \xrightarrow {\epsilon \otimes \operatorname{id}_{F(X)} } F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes F(X) \xrightarrow {\mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, X} } F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes X)$

is an isomorphism for each $X \in \operatorname{\mathcal{C}}$.

Example 2.1.5.14. In the special case where $\operatorname{\mathcal{C}}= \{ e \}$, we can identify a nonunital lax monoidal functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with a nonunital algebra object $A$ of $\operatorname{\mathcal{D}}$. In this case, Proposition 2.1.5.13 asserts that a morphism $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow A$ is a left unit (in the sense of Definition 2.1.5.1) if and only if the diagram

$\xymatrix@R =50pt@C=50pt{ \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{\epsilon \otimes \epsilon } \ar [d]^{\upsilon } & A \otimes A \ar [d]^{m} \\ \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{\epsilon } & A }$

is commutative (that is, $\epsilon$ is idempotent) and the map

$\mathbf{1}_{\operatorname{\mathcal{D}}} \otimes A \xrightarrow {\epsilon \otimes \operatorname{id}_{A} } A \otimes A \xrightarrow {m} A$

is a monomorphism in $\operatorname{\mathcal{D}}$ (that is, $\epsilon$ is left cancellative). When $\operatorname{\mathcal{D}}$ is the category of sets (equipped with the cartesian monoidal structure of Example 2.1.3.2), this reduces to the statement of Proposition 2.1.2.3.

Proof of Proposition 2.1.5.13. To simplify the notation, let us use the symbol $\mathbf{1}$ to denote the unit objects of both $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$ for the unit constraints of both $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, and $\lambda$ for the unit constraints of both $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor equipped with a nonunital lax monoidal structure $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$. Suppose first that $\epsilon : \mathbf{1} \rightarrow F( \mathbf{1} )$ is a left unit for $F$. Then the diagram

$\xymatrix@C =50pt{ \mathbf{1} \otimes \mathbf{1} \ar [dd]_{ \lambda _{\mathbf{1}}} \ar [r]^-{ \operatorname{id}_{ \mathbf{1}} \otimes \epsilon } & \mathbf{1} \otimes F( \mathbf{1} ) \ar [r]^-{ \epsilon \otimes \operatorname{id}_{ F( \mathbf{1})} } \ar [ddr]_-{ \lambda _{ F(\mathbf{1})}} & F(\mathbf{1}) \otimes F(\mathbf{1}) \ar [d]^{ \mu _{ \mathbf{1}, \mathbf{1}}} \\ & & F( \mathbf{1} \otimes \mathbf{1} ) \ar [d]^{ F( \lambda _{\mathbf{1}} ) } \\ \mathbf{1} \ar [rr]^{\epsilon } & & F( \mathbf{1} ) }$

commutes: the region on the left commutes by the naturality of the left unit constraints for $\operatorname{\mathcal{D}}$ (Remark 2.1.2.18), and the region on the right commutes by virtue of our assumption that $\epsilon$ is a left unit. The commutativity of the outer square shows that $\epsilon$ satisfies condition $(1)$ of Proposition 2.1.5.13 (by virtue of the fact that the unit constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are given by $\upsilon = \lambda _\mathbf {1}$; see Corollary 2.1.2.21). For every object $X \in \operatorname{\mathcal{C}}$, the composition

$\mathbf{1} \otimes F(X) \xrightarrow { \epsilon \otimes \operatorname{id}_{F(X)} } F( \mathbf{1} ) \otimes F(X) \xrightarrow { \mu _{ \mathbf{1}, X} } F( \mathbf{1} \otimes X) \xrightarrow { F( \lambda _{X}) } F(X)$

is the left unit constraint $\lambda _{F(X)}$, which is an isomorphism. Since $F( \lambda _{X} )$ is also an isomorphism, it follows that the composition $\mu _{ \mathbf{1}, X} \circ ( \epsilon \otimes \operatorname{id}_{F(X)} )$ is an isomorphism.

Now suppose that $\epsilon$ satisfies conditions $(1)$ and $(2)$; we wish to show that it is a left unit for $F$. Fix an object $X \in \operatorname{\mathcal{C}}$, and let $f: \mathbf{1} \otimes F(X) \rightarrow F(X)$ denote the composition

$\mathbf{1} \otimes F(X) \xrightarrow { \epsilon \otimes \operatorname{id}_{F(X)}} F( \mathbf{1} ) \otimes F(X) \xrightarrow { \mu _{ \mathbf{1}, X} } F( \mathbf{1} \otimes X) \xrightarrow { F( \lambda _ X) } F(X).$

We wish to show that $f$ is equal to the left unit constraint $\lambda _{F(X)}$ for the monoidal category $\operatorname{\mathcal{D}}$. Unwinding the definitions, this is equivalent to the assertion that $\operatorname{id}_{ \mathbf{1} } \otimes f$ is equal to the composition

$\mathbf{1} \otimes (\mathbf{1} \otimes F(X) ) \xrightarrow { \alpha _{ \mathbf{1}, \mathbf{1}, X} } ( \mathbf{1} \otimes \mathbf{1}) \otimes F(X) \xrightarrow { \upsilon \otimes \operatorname{id}_{F(X)} } \mathbf{1} \otimes F(X).$

By virtue of assumption $(2)$, it will suffice to prove that these morphisms agree after postcomposition with the monomorphism

$\mathbf{1} \otimes F(X) \xrightarrow { \epsilon \otimes \operatorname{id}_{F(X)} } F( \mathbf{1} ) \otimes F(X) \xrightarrow { \mu _{ \mathbf{1}, X} } F( \mathbf{1} \otimes X).$

This is equivalent to the commutativity of the outer rectangle in the diagram

$\xymatrix@R =50pt@C=24pt{ \mathbf{1} \otimes (\mathbf{1} \otimes F(X) ) \ar [r]^-{\epsilon } \ar [dr]^-{ \epsilon \otimes \epsilon } \ar [dd]^{\alpha } & \mathbf{1} \otimes ( F(\mathbf{1}) \otimes F(X) ) \ar [d]^{\epsilon } \ar [r]^-{ \mu } & \mathbf{1} \otimes F( \mathbf{1} \otimes X) \ar [d]^{\epsilon } \ar [r]^-{ F( \lambda _ X)} & \mathbf{1} \otimes F(X) \ar [d]^{\epsilon } \\ & F( \mathbf{1}) \otimes ( F(\mathbf{1}) \otimes F(X)) \ar [r]^-{\mu } \ar [d]^{\alpha } & F(\mathbf{1}) \otimes F(\mathbf{1} \otimes X) \ar [r]^-{F( \lambda _ X)} \ar [d]^{\mu } & F( \mathbf{1} ) \otimes F(X) \ar [d]^{\mu } \\ (\mathbf{1} \otimes \mathbf{1} ) \otimes F(X) \ar [r]^-{ \epsilon \otimes \epsilon } \ar [dd]^{\upsilon } & ( F(\mathbf{1}) \otimes F(\mathbf{1})) \otimes F(X) \ar [d]^{\mu } & F( \mathbf{1} \otimes (\mathbf{1} \otimes X) ) \ar [d]^{F(\alpha )} \ar [r]^-{ F( \operatorname{id}\otimes \lambda _ X) } & F(\mathbf{1} \otimes X) \ar [dd]^{\operatorname{id}} \\ & F( \mathbf{1} \otimes \mathbf{1}) \otimes F(X) \ar [r]^-{\mu } \ar [d]^{ F( \upsilon ) } & F( (\mathbf{1} \otimes \mathbf{1}) \otimes X) \ar [dr]^-{ F( \upsilon \otimes \operatorname{id})} & \\ \mathbf{1} \otimes F(X) \ar [r]^-{ \epsilon } & F( \mathbf{1} ) \otimes F(X) \ar [rr]^-{\mu } & & F( \mathbf{1} \otimes X). }$

In fact, the whole diagram commutes: the rectangle on the lower left commutes by virtue of our assumption that $\epsilon$ satisfies $(1)$, the rectangle in the middle commutes by virtue of the compatibility of the $\mu$ with the associativity constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, the square on the lower right commutes by the construction of the left unit constraint $\lambda _{X}$, and the remaining regions commute by naturality. $\square$

Example 2.1.5.15. Let $k$ be a field, let $\operatorname{Vect}_{k}$ denote the category of vector spaces over $k$, and let $F: \operatorname{Vect}_{k} \rightarrow \operatorname{Set}$ be the forgetful functor, endowed with the nonunital lax monoidal structure described in Example 2.1.4.5. Then $F$ is a lax monoidal functor: the function

$\epsilon : \{ \ast \} \rightarrow F(k) \quad \quad \epsilon (\ast ) = 1 \in k$

is a left and right unit for $F$.

Example 2.1.5.15 illustrates a special case of a general phenomenon:

Example 2.1.5.16. Let $\operatorname{\mathcal{C}}$ be a monoidal category, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor corepresented by the unit object $\mathbf{1} \in \operatorname{\mathcal{C}}$, given concretely by the formula $F(X ) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \mathbf{1}, X )$. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have a canonical map

$\mu _{X,Y}: F(X) \times F(Y) \rightarrow F( X \otimes Y ),$

which carries a pair of elements $x \in F(X)$, $y \in F(Y)$ to the composite map

$\mathbf{1} \xrightarrow { \upsilon ^{-1} } \mathbf{1} \otimes \mathbf{1} \xrightarrow {x \otimes y} X \otimes Y.$

The collection of maps $\{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ determines a lax monoidal structure on the functor $F$, with unit given by the map

$\epsilon : \{ \ast \} \rightarrow F(\mathbf{1}) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \mathbf{1}, \mathbf{1} ) \quad \quad \epsilon (\ast ) = \operatorname{id}_{ \mathbf{1} }.$

Example 2.1.5.17. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories which admit finite products, and regard $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ as endowed with the cartesian monoidal structures described in Example 2.1.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be any functor, and let $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ be the induced functor of opposite categories. Then the functor $F^{\operatorname{op}}$ admits a lax monoidal structure, which associates to each pair of objects $X,Y \in \operatorname{\mathcal{C}}$ the canonical map $\mu _{X,Y}: F(X \times Y) \rightarrow F(X) \times F(Y)$ in the category $\operatorname{\mathcal{D}}$ (which we can view as a morphism from $F^{\operatorname{op}}(X) \otimes F^{\operatorname{op}}(Y) \rightarrow F^{\operatorname{op}}(X \otimes Y)$ in the category $\operatorname{\mathcal{D}}^{\operatorname{op}}$). The unit for $F$ is given by the unique morphism $\epsilon : F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \rightarrow \mathbf{1}_{\operatorname{\mathcal{D}}}$ in the category $\operatorname{\mathcal{D}}$ (where $\mathbf{1}_{\operatorname{\mathcal{C}}}$ and $\mathbf{1}_{\operatorname{\mathcal{D}}}$ are final objects of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively).

Definition 2.1.5.18. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be lax monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. We will say that a natural transformation $\gamma : F \rightarrow F'$ is monoidal if it satisfies the following pair of conditions:

• The natural transformation $\gamma$ is nonunital monoidal, in the sense of Definition 2.1.4.10. That is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the diagram

$\xymatrix@C =50pt@R=50pt{ F(X) \otimes F(Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ \gamma (X) \otimes \gamma (Y) } & F(X \otimes Y) \ar [d]^{\gamma (X \otimes Y)} \\ F'(X) \otimes F'(Y) \ar [r]^-{ \mu '_{X,Y}} & F'(X \otimes Y) }$

commutes, where $\mu$ and $\mu '$ are the tensor constraints of $F$ and $F'$, respectively.

• The unit of $F'$ is equal to the composition $\mathbf{1}_{\operatorname{\mathcal{D}}} \xrightarrow {\epsilon } F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \xrightarrow { \gamma ( \mathbf{1}_{\operatorname{\mathcal{C}}} )} F'(\mathbf{1}_{\operatorname{\mathcal{C}}} )$, where $\epsilon$ is the unit of $F$.

We let $\operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the category whose objects are lax monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and whose morphisms are monoidal natural transformations, which we regard as a (non-full) subcategory of the category $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ introduced in Definition 2.1.4.10.

Remark 2.1.5.19 (Compatibility with Reversal). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor, and let $F^{\operatorname{rev}}: \operatorname{\mathcal{C}}^{\operatorname{rev}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{rev}}$ be as in Example 2.1.4.13. Then $F$ is a lax monoidal functor if and only if $F^{\operatorname{rev}}$ is a lax monoidal functor. This observation (and its counterpart for monoidal natural transformations) supplies an isomorphism of categories $\operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{C}}^{\operatorname{rev}}, \operatorname{\mathcal{D}}^{\operatorname{rev}} )$.

Remark 2.1.5.20 (Closure under Composition). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors equipped with nonunital lax monoidal structures $\mu$ and $\nu$, respectively, so that the composite functor $G \circ F$ inherits a nonunital lax monoidal structure (Construction 2.1.4.17). If $F$ and $G$ admit units

$\delta : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \quad \quad \epsilon : \mathbf{1}_{\operatorname{\mathcal{E}}} \rightarrow G( \mathbf{1}_{\operatorname{\mathcal{D}}} ),$

then the composite map

$\mathbf{1}_{\operatorname{\mathcal{E}}} \xrightarrow { \epsilon } G( \mathbf{1}_{\operatorname{\mathcal{D}}} ) \xrightarrow { G( \delta )} (G \circ F)( \mathbf{1}_{\operatorname{\mathcal{C}}} )$

is a unit for the composite functor $G \circ F$. This observation (and its counterpart for monoidal natural transformations) imply that the composition law of Construction 2.1.4.17 restricts to a functor

$\circ : \operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}).$

Example 2.1.5.21 (Algebra Objects). Let $\operatorname{\mathcal{C}}$ be a monoidal category. An algebra object of $\operatorname{\mathcal{C}}$ is a pair $(A,m)$, where $A$ is an object of $\operatorname{\mathcal{C}}$ and $m: A \otimes A \rightarrow A$ is a nonunital algebra structure on $A$ (Example 2.1.4.11) for which there exists a unit $\epsilon : \mathbf{1} \rightarrow A$ (in the sense of Definition 2.1.5.1). If $(A,m)$ and $(A',m')$ are algebra objects of $\operatorname{\mathcal{C}}$ with units $\epsilon : \mathbf{1} \rightarrow A$ and $\epsilon ': \mathbf{1} \rightarrow A'$, then we say that a morphism $f: A \rightarrow A'$ is an algebra homomorphism if it is a nonunital algebra homomorphism (Example 2.1.4.11) which satisfies $\epsilon ' = f \circ \epsilon$. We let $\operatorname{Alg}(\operatorname{\mathcal{C}})$ denote the category whose objects are algebra objects of $\operatorname{\mathcal{C}}$ and whose morphisms are algebra homomorphisms. We regard $\operatorname{Alg}(\operatorname{\mathcal{C}})$ as a (non-full) subcategory of the category $\operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$ of nonunital algebra objects of $\operatorname{\mathcal{C}}$ defined in Example 2.1.4.11.

Let $\{ e\}$ denote the trivial monoid, regarded as a (strict) monoidal category having only identity morphisms (Example 2.1.1.3). Then algebra objects of $\operatorname{\mathcal{C}}$ can be identified with lax monoidal functors $\{ e\} \rightarrow \operatorname{\mathcal{C}}$. More precisely, the isomorphism $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \{ e \} , \operatorname{\mathcal{C}}) \simeq \operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$ of Example 2.1.4.11 specializes to an isomorphism of (non-full) subcategories $\operatorname{Fun}^{\operatorname{lax}}( \{ e\} , \operatorname{\mathcal{C}}) \simeq \operatorname{Alg}(\operatorname{\mathcal{C}})$.

Example 2.1.5.22. Let $\operatorname{Set}$ denote the category of sets, equipped with the cartesian monoidal structure of Example 2.1.3.2. Then we can identify algebra objects of $\operatorname{Set}$ with monoids. More precisely, there is a canonical isomorphism of categories $\operatorname{Alg}(\operatorname{Set}) \simeq \operatorname{Mon}$, where $\operatorname{Mon}$ denotes the category of monoids (Definition 1.3.2.3).

For later use, we record the following elementary fact about algebra objects of a monoidal category $\operatorname{\mathcal{C}}$:

Proposition 2.1.5.23. Let $\operatorname{\mathcal{C}}$ be a monoidal category and let $(A,m)$ be an algebra object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The unit map $\epsilon : \mathbf{1} \rightarrow A$ is an isomorphism in $\operatorname{\mathcal{C}}$.

$(2)$

The object $A$ is invertible: that is, there exists an object $B \in \operatorname{\mathcal{C}}$ for which the tensor products $A \otimes B$ and $B \otimes A$ are isomorphic to $\mathbf{1}$.

$(3)$

The construction $X \mapsto A \otimes X$ determines a fully faithful functor from $\operatorname{\mathcal{C}}$ to itself.

Proof. The implications $(1) \Rightarrow (2) \Rightarrow (3)$ are immediate. We will prove that $(3)$ implies $(1)$. It follows from assumption that $(3)$ that there is a unique morphism $f: A \rightarrow \mathbf{1}$ for which the lower right triangle in the diagram

$\xymatrix@R =50pt@C=50pt{ A \otimes \mathbf{1} \ar [r]^-{\operatorname{id}_ A \otimes \epsilon } \ar [d]^{\rho _{A}} & A \otimes A \ar [d]^{\operatorname{id}_ A \otimes f} \ar [dl]_{m} \\ A \ar [r]^{ \rho _{A}^{-1} } & A \otimes \mathbf{1} }$

commutes. The upper left triangle also commutes, since $\epsilon$ is a right unit with respect to the multiplication $m$. It follows that the square commutes: that is, the composition

$A \otimes \mathbf{1} \xrightarrow { \operatorname{id}_{A} \otimes \epsilon } A \otimes A \xrightarrow { \operatorname{id}_{A} \otimes f } A \otimes \mathbf{1}$

is equal to the identity. Invoking assumption $(3)$, we conclude that $f$ is a left inverse to $\epsilon$: that is, the composition $f \circ \epsilon$ is equal to the identity on the unit object $\mathbf{1}$.

We now show that $f$ is also a right inverse to $\epsilon$: that is, the composition $\epsilon \circ f$ is equal to the identity morphism $\operatorname{id}_{A}$. Consider the diagram

$\xymatrix@R =50pt@C=50pt{ A \ar [r]^{ \lambda _{A}^{-1} } \ar [dd]^{f} & \mathbf{1} \otimes A \ar [r]^{ \epsilon \otimes \operatorname{id}_{A} } \ar [d]^{ \operatorname{id}_{\mathbf{1}} \otimes f } & A \otimes A \ar [d]^{ \operatorname{id}_{A} \otimes f} \\ & \mathbf{1} \otimes \mathbf{1} \ar [r]^{ \epsilon \otimes \operatorname{id}_{\mathbf{1} } } \ar [dl]_{\upsilon } & A \otimes \mathbf{1} \ar [d]^{ \rho _{A} } \\ \mathbf{1} \ar [rr]^{\epsilon } & & A. }$

The defining property of $f$ guarantees that the vertical composition on the right coincides with the multiplication map $m: A \otimes A \rightarrow A$. The assumption that $\epsilon$ is a left unit with respect to the multiplication $m$ shows that clockwise composition around the diagram gives the identity map $\operatorname{id}_{A}: A \rightarrow A$. To complete the proof, it will suffice to show that the diagram commutes. The commutativity of the upper right square follows from the functoriality of the tensor product, the commutativity of the trapezoidal region on the left follows from the functoriality of the left unit constraints of $\operatorname{\mathcal{C}}$, and the commutativity of the trapezoidal region on the bottom from the functoriality of the right unit constraints of $\operatorname{\mathcal{C}}$ (here we invoke the fact that the map $\upsilon : \mathbf{1} \otimes \mathbf{1} \xrightarrow {\sim } \mathbf{1}$ coincides with both $\lambda _{\mathbf{1}}$ and $\rho _{\mathbf{1}}$; see Corollary 2.1.2.21). $\square$