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 $\epsilon _{L} = \epsilon _{R}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
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 $\epsilon _{R} = \epsilon _{L}$. $\square$