Remark 2.1.6.4 (00E1)

Remark 2.1.6.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital monoidal functor. Let $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$ be an isomorphism in the category $\operatorname{\mathcal{C}}$. Then $\epsilon $ automatically satisfies condition $(2)$ of Proposition 2.1.5.13: for each $X \in \operatorname{\mathcal{C}}$, both of the maps

\[ \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes F(X) \xrightarrow { \epsilon \otimes \operatorname{id}_{F(X)} } F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes F(X) \xrightarrow { \mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, X} } F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes X) \]

are isomorphisms. It follows that $\epsilon $ is a unit for $F$ if and only if it satisfies condition $(1)$ of Proposition 2.1.5.13: that is, if and only if the diagram

\[ \xymatrix@R =50pt@C=50pt{ \mathbf{1}_{\operatorname{\mathcal{D}}} \otimes \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{ \epsilon \otimes \epsilon } \ar [dd]^{ \upsilon _{\operatorname{\mathcal{D}}} } & F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \ar [d]^{ \mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, \mathbf{1}_{\operatorname{\mathcal{C}}} }} \\ & F( \mathbf{1}_{\operatorname{\mathcal{C}}} \otimes \mathbf{1}_{\operatorname{\mathcal{C}}}) \ar [d]^{ F( \upsilon _{\operatorname{\mathcal{C}}}) } \\ \mathbf{1}_{\operatorname{\mathcal{D}}} \ar [r]^-{ \epsilon } & F( \mathbf{1}_{\operatorname{\mathcal{C}}}) } \]

is commutative. By virtue of Proposition 2.1.2.9, there exists an isomorphism $\epsilon $ satisfying this condition if and only if the pair $( F( \mathbf{1}_{\operatorname{\mathcal{C}}} ), F( \upsilon _{\operatorname{\mathcal{C}}}) \circ \mu _{ \mathbf{1}_{\operatorname{\mathcal{C}}}, \mathbf{1}_{\operatorname{\mathcal{C}}}})$ is a unit of $\operatorname{\mathcal{C}}$ (in the sense of Definition 2.1.2.5).

In other words, a nonunital monoidal functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is monoidal if and only if the functors

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}\quad \quad X \mapsto F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \otimes X \]

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}\quad \quad X \mapsto X \otimes F( \mathbf{1}_{\operatorname{\mathcal{C}}} ) \]

are fully faithful (in which case they are both canonically isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{D}}$).