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 $X \in \operatorname{\mathcal{C}}$.
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.
$\square$