$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
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 $G$.
$\square$