Definition 2.1.7.10. Let $\operatorname{\mathcal{A}}$ be a monoidal category, and let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\operatorname{\mathcal{A}}$-enriched categories. An $\operatorname{\mathcal{A}}$-enriched functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ consists of the following data:
- $(1)$
For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, and object $F(X) \in \operatorname{Ob}(\operatorname{\mathcal{D}})$.
- $(2)$
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, a morphism
\[ F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]in the category $\operatorname{\mathcal{A}}$.
These data are required to satisfy the following conditions:
For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the morphism $e_{F(X)}: \mathbf{1} \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(X) )$ factors as a composition
\[ \mathbf{1} \xrightarrow { e_{X} } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, X) \xrightarrow { F_{X,X} } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(X) ). \]For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the diagram
\[ \xymatrix@R =50pt@C=50pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r] \ar [d]^{ F_{Y,Z} \otimes F_{X,Y} } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{ F_{X,Z} } \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(Y), F(Z) ) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \ar [r] & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) } \]commutes (in the category $\operatorname{\mathcal{A}}$); here the horizontal maps are given by the composition laws on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$.