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2.1.4 Nonunital Monoidal Functors

We now study functors between (nonunital) monoidal categories.

Definition 2.1.4.1 (Strict Nonunital Monoidal Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories (Definition 2.1.1.5). A nonunital strict monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with the following properties:

  • The diagram of functors

    \[ \xymatrix { \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\ar [r]^-{\otimes } \ar [d]^{F \times F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}\times \operatorname{\mathcal{D}}\ar [r]^-{ \otimes } & \operatorname{\mathcal{D}}} \]

    is strictly commutative. In particular, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have an equality $F(X) \otimes F(Y) = F(X \otimes Y)$ of objects of $\operatorname{\mathcal{D}}$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the functor $F$ carries the associativity constraint $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ (for the monoidal structure on $\operatorname{\mathcal{C}}$) to the associativity constraint $\alpha _{F(X), F(Y), F(Z) }: F(X) \otimes (F(Y) \otimes F(Z) ) \simeq (F(X) \otimes F(Y) ) \otimes F(Z)$ (for the monoidal structure on $\operatorname{\mathcal{D}}$).

Example 2.1.4.2. Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category. Then the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ is a nonunital strict monoidal functor from $\operatorname{\mathcal{C}}$ to itself.

For many applications, Definition 2.1.4.1 is too restrictive. In practice, the definition of a (nonunital) monoidal structure $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ on a category $\operatorname{\mathcal{C}}$ often involves constructions which are only well-defined up to isomorphism (see Examples 2.1.3.1 and 2.1.3.2). In such cases, it is unreasonable to require that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ has the property that $F(X) \otimes F(Y)$ and $F(X \otimes Y)$ are the same object of $\operatorname{\mathcal{D}}$. Instead, we should ask for any isomorphism $\mu _{X,Y}: F(X) \otimes F(Y) \xrightarrow {\sim } F(X \otimes Y)$. To get a well-behaved theory, we should further demand that the isomorphisms $\mu _{X,Y}$ depend functorially on $X$ and $Y$, and are suitable compatible with the associativity constraints on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. We begin by considering a slightly more general situation, where the morphisms $\mu _{X,Y}$ are not required to be invertible.

Definition 2.1.4.3 (Nonunital Lax Monoidal Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. A nonunital lax monoidal structure on $F$ is a collection of morphisms $\mu = \{ \mu _{X,Y}: F(X) \otimes F(Y) \xrightarrow {\sim } F(X \otimes Y) \} _{X,Y \in \operatorname{\mathcal{C}}}$ which satisfy the following pair of conditions:

$(a)$

The morphisms $\mu _{X,Y}$ depend functorially on $X$ and $Y$: that is, for every pair of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$ in $\operatorname{\mathcal{C}}$, the diagram

\[ \xymatrix { F(X) \otimes F(Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ F(f) \otimes F(g) } & F( X \otimes Y) \ar [d]^{ F( f \otimes g) } \\ F(X') \otimes F(Y' ) \ar [r]^-{ \mu _{X',Y'} } & F( X' \otimes Y' ) } \]

commutes (in the category $\operatorname{\mathcal{D}}$). In other words, we can regard $\mu $ as a natural transformation of functors as indicated in the diagram

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\ar [r]^-{\otimes } \ar [d]_{F \times F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}\times \operatorname{\mathcal{D}}\ar [r]^-{ \otimes } \ar@ {=>}[]+<20pt,20pt>;+<60pt,50pt>_-{\mu } & \operatorname{\mathcal{D}}. } \]
$(b)$

The morphisms $\mu _{X,Y}$ are compatible with the associativity constraints on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ in the following sense: for every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram

\[ \xymatrix@R =40pt@C=70pt{ F(X) \otimes (F(Y) \otimes F(Z) ) \ar [r]^-{ \alpha _{F(X), F(Y), F(Z) } } \ar [d]^{ \operatorname{id}_{F(X)} \otimes \mu _{Y,Z} } & (F(X) \otimes F(Y) ) \otimes F(Z) \ar [d]^{ \mu _{X,Y} \otimes \operatorname{id}_{F(Z)}} \\ F(X) \otimes F(Y \otimes Z) \ar [d]^{ \mu _{X,Y \otimes Z} } & F(X \otimes Y) \otimes F(Z) \ar [d]^{ \mu _{X \otimes Y, Z}} \\ F( X \otimes (Y \otimes Z) ) \ar [r]^{ F( \alpha _{X,Y,Z } )} & F( (X \otimes Y) \otimes Z) } \]

commutes (in the category $\operatorname{\mathcal{D}}$).

A nonunital 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 a functor and $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ is a nonunital lax monoidal structure on $F$. In this case, we will refer to the isomorphisms $\{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ as the tensor constraints of $F$.

Definition 2.1.4.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. A nonunital monoidal structure on $F$ is a lax nonunital monoidal structure $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ on $F$ with the property that each of the tensor constraints $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y)$ is an isomorphism.

A nonunital 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 a functor and $\mu $ is a nonunital monoidal structure on $F$.

Example 2.1.4.5. Let $k$ be a field and let $\operatorname{Vect}_{k}$ be the category of vector spaces over $k$, endowed with the monoidal structure of Example 2.1.3.1. The construction of this monoidal structure involved certain choices: for every pair of vector spaces $V,W \in \operatorname{Vect}_{k}$, we selected a universal bilinear map $b_{V,W}: V \times W \rightarrow V \otimes _{k} W$. The collection of functions $b = \{ b_{V,W} \} _{V,W \in \operatorname{Vect}_{k} }$ is then a nonunital lax monoidal structure on the forgetful functor $\operatorname{Vect}_{k} \rightarrow \operatorname{Set}$ (where we equip $\operatorname{Set}$ with the monoidal structure given by Cartesian products; see Example 2.1.3.2). Note that the tensor product functor $\otimes _{k}: \operatorname{Vect}_{k} \times \operatorname{Vect}_ k \rightarrow \operatorname{Vect}_{k}$ is characterized by the requirement that it is given on objects by $(V,W) \mapsto V \otimes _{k} W$ and satisfies condition $(a)$ of Definition 2.1.4.3, and the associativity constraint on $\operatorname{Vect}_{k}$ is characterized by the requirement that it satisfies condition $(b)$ of Definition 2.1.4.3. Note that $b$ is not a nonunital monoidal structure: the bilinear maps $b_{V,W}: V \times W \rightarrow V \otimes _{k} W$ are never bijective, except in the trivial case where $V \simeq 0 \simeq W$.

Example 2.1.4.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strict nonunital monoidal functor. Then $F$ admits a nonunital monoidal structure $\{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$, where we take each $\mu _{X,Y}$ to be the identity morphism from $F(X) \otimes F(Y) = F(X \otimes Y)$ to itself.

Conversely, if $(F, \mu )$ is a nonunital monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ with the property that the tensor constraints $\mu _{X,Y}$ is an identity morphism in $\operatorname{\mathcal{D}}$, then $F$ is a strict nonunital monoidal functor.

Example 2.1.4.7. Let $M$ and $M'$ be nonunital monoids, regarded as nonunital monoidal categories having only identity morphisms (Example 2.1.1.3). Then nonunital lax monoidal functors from $M$ to $M'$ (in the sense of Definition 2.1.4.3) can be identified with nonunital monoid homomorphisms from $M$ to $M'$ (in the sense of Definition 2.1.0.5). Moreover, every nonunital lax monoidal functor from $M$ to $M'$ is automatically strict.

Example 2.1.4.8 (The Left Regular Representation). Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category and let $\operatorname{End}(\operatorname{\mathcal{C}}) = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ be the category of functors from $\operatorname{\mathcal{C}}$ to itself, endowed with the strict monoidal structure of Example 2.1.1.4. For each object $X \in \operatorname{\mathcal{C}}$, let $\ell _{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denote the functor given on objects by the formula $\ell _{X}(Y) = X \otimes Y$. The construction $X \mapsto \ell _{X}$ then determines a functor $\ell : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there is a natural isomorphism $\mu _{X,Y}: \ell _{X} \circ \ell _{Y} \xrightarrow {\sim } \ell _{X \otimes Y}$, whose value on an object $Z \in \operatorname{\mathcal{C}}$ is given by the associativity constraint

\[ (\ell _{X} \circ \ell _{Y})(Z) = X \otimes (Y \otimes Z) \xrightarrow { \alpha _{X,Y,Z} } (X \otimes Y) \otimes Z = \ell _{X \otimes Y}(Z). \]

Then $\mu = \{ \mu _{X,Y} \} _{X,Y}$ is a nonunital monoidal structure on the functor $X \mapsto \ell _{X}$: property $(a)$ of Definition 2.1.4.3 follows from the naturality of the associativity constraint on $\operatorname{\mathcal{C}}$, and property $(b)$ is a reformulation of the pentagon identity.

Warning 2.1.4.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories. A strict nonunital monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ possessing certain properties. However, a nonunital (lax) monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ together with additional structure, given by the tensor constraints $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y)$. We will often abuse terminology by identifying a nonunital (lax) monoidal functor $(F, \mu )$ with the underlying functor $F$; in this case, we implicitly assume that the tensor constraints $\mu _{X,Y}$ have been specified.

Definition 2.1.4.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories. Let $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors equipped with nonunital lax monoidal structures $\mu $ and $\mu '$, respectively. We say that a natural transformation of functors $\gamma : F \rightarrow F'$ is nonunital monoidal if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the diagram

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

is commutative.

We let $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the category whose objects are nonunital lax monoidal functors $(F,\mu )$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$, and whose morphisms are nonunital monoidal natural transformations, and we let $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the the nonunital monoidal functors $(F,\mu )$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.

Example 2.1.4.11 (Nonunital Algebras). Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category and let $A$ be an object of $\operatorname{\mathcal{C}}$. A nonunital algebra structure on $A$ is a map $m: A \otimes A \rightarrow A$ for which the diagram

\[ \xymatrix { & A \otimes (A \otimes A) \ar [rr]_-{\alpha _{A,A,A}} \ar [dl]_{ \operatorname{id}\otimes m} & & (A \otimes A) \otimes A \ar [dr]^{m \otimes \operatorname{id}} & \\ A \otimes A \ar [drr]^{m} & & & & A \otimes A \ar [dll]_{m} \\ & & A & & } \]

is commutative. A nonunital algebra object of $\operatorname{\mathcal{C}}$ is a pair $(A,m)$, where $A$ is an object of $\operatorname{\mathcal{C}}$ and $m$ is a nonunital algebra structure on $A$. If $(A,m)$ and $(A', m')$ are nonunital algebra objects of $\operatorname{\mathcal{C}}$, then we say that a morphism $f: A \rightarrow A'$ is a nonunital algebra homomorphism if the diagram

\[ \xymatrix { A \otimes A \ar [r]^{m} \ar [d]^{ f \otimes f} & A \ar [d]^{f} \\ A' \otimes A' \ar [r]^{m'} & A' } \]

is commutative. We let $\operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$ denote the category whose objects are nonunital algebra objects of $\operatorname{\mathcal{C}}$ and whose morphisms are nonunital algebra homomorphisms.

Let $\{ e\} $ denote the trivial monoid, regarded as a (strict) monoidal category having only identity morphisms (Example 2.1.1.3). Then we can identify objects $A \in \operatorname{\mathcal{C}}$ with functors $F: \{ e\} \rightarrow \operatorname{\mathcal{C}}$ (by means of the formula $A = F(e)$). Unwinding the definitions, we see that nonunital lax monoidal structures on the functor $F$ (in the sense of Definition 2.1.4.3) can be identified with nonunital algebra structures on the object $A = F(e)$. Under this identification, nonunital monoidal natural transformations correspond to homomorphisms of nonunital algebras. We therefore have an isomorphism of categories $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \{ e \} , \operatorname{\mathcal{C}}) \simeq \operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$.

Example 2.1.4.12. Let $\operatorname{Set}$ denote the category of sets, endowed with the monoidal structure given by Cartesian product of sets (Example 2.1.3.2). For each set $S$, we can identify nonunital algebra structures on $S$ (in the sense of Example 2.1.4.11) with nonunital monoid structures on $S$ (in the sense of Definition 2.1.0.3). This observation supplies an isomorphism of categories $\operatorname{Alg}^{\operatorname{nu}}( \operatorname{Set}) \simeq \operatorname{Mon}^{\operatorname{nu}}$, where $\operatorname{Mon}^{\operatorname{nu}}$ is the category of Definition 2.1.0.5.

Example 2.1.4.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ and $\operatorname{\mathcal{D}}^{\operatorname{rev}}$ denote the same categories with the reversed nonunital monoidal structure (Example 2.1.3.5). Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ can be also regarded as a functor from $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ to $\operatorname{\mathcal{D}}^{\operatorname{rev}}$, which we will denote by $F^{\operatorname{rev}}$. There is a canonical bijection

\[ \{ \text{Nonunital lax monoidal structures on $F$} \} \simeq \{ \text{Nonunital lax monoidal structures on $F^{\operatorname{rev}}$} \} , \]

which carries a nonunital lax monoidal structure $\mu $ to the the nonunital lax monoidal structure $\mu ^{\operatorname{rev}}$ given by the formula $\mu ^{\operatorname{rev}}_{X,Y} = \mu _{Y,X}$. Using these bijections, we obtain a canonical isomorphism of categories $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{rev}}, \operatorname{\mathcal{D}}^{\operatorname{rev}} )$, which restricts to an isomorphism $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{rev}}, \operatorname{\mathcal{D}}^{\operatorname{rev}} )$.

Example 2.1.4.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and regard the opposite categories $\operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\operatorname{\mathcal{D}}^{\operatorname{op}}$ as equipped with the nonunital monoidal structures of Example 2.1.3.4. Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ determines a functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$. There is a canonical bijection

\[ \{ \text{Nonunital monoidal structures on $F$} \} \simeq \{ \text{Nonunital monoidal structures on $F^{\operatorname{op}}$} \} , \]

which carries a nonunital monoidal structure $\mu $ on $F$ to a nonunital monoidal structure $\mu '$ on $F^{\operatorname{op}}$, given concretely by $\mu '_{X,Y} = \mu _{X,Y}^{-1}$. Using these bijections, we obtain a canonical isomorphism of categories $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \simeq \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} )$.

Warning 2.1.4.15. The analogue of Example 2.1.4.14 for nonunital lax monoidal functors is false. The notion of nonunital lax monoidal functor is not self-opposite: in general, there is no simple relationship between the categories $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ and $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} )$.

Construction 2.1.4.16 (Composition of Nonunital Monoidal Functors). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be nonunital monoidal categories, and suppose we are given a pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. If $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ is a nonunital lax monoidal structure on the functor $F$ and $\nu = \{ \nu _{U,V} \} _{U, V \in \operatorname{\mathcal{D}}}$ is a nonunital lax monoidal structure on $G$, then the composite functor $G \circ F$ inherits a nonunital lax monoidal structure, which associates to each pair of objects $X,Y \in \operatorname{\mathcal{C}}$ the composite map

\[ (G \circ F)(X) \otimes (G \circ F)(Y) \xrightarrow { \nu _{ F(X), F(Y)} } G( F(X) \otimes F(Y) ) \xrightarrow { F( \mu _{X,Y} ) } (G \circ F)(X \otimes Y). \]

This construction determines a composition law

\[ \circ : \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). \]

Remark 2.1.4.17. In the situation of Construction 2.1.4.16, suppose that $\mu $ and $\nu $ are nonunital monoidal structures on $F$ and $G$, respectively: that is, assume that all of the the tensor constraints

\[ \mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F( X \otimes Y) \quad \quad \nu _{U,V}: G(U) \otimes G(V) \rightarrow G( U \otimes V) \]

are isomorphisms. Then Construction 2.1.4.16 supplies a nonunital monoidal structure on the composite functor $G \circ F$. We therefore obtain a composition law

\[ \circ : \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). \]