Proposition 2.1.5.6 (00MS)

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. $\square$