Remark 2.1.7.4 (Functoriality). Let $\operatorname{\mathcal{A}}$ and $\operatorname{\mathcal{A}}'$ be monoidal categories, and let $F: \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{A}}'$ be a lax monoidal functor (with tensor constraints $\mu _{A,B}: F(A) \otimes F(B) \rightarrow F(A \otimes B)$ and unit $\epsilon : \mathbf{1}_{ \operatorname{\mathcal{A}}' } \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{A}}} )$). Then every $\operatorname{\mathcal{A}}$-enriched category $\operatorname{\mathcal{C}}$ determines an $\operatorname{\mathcal{A}}'$-enriched category $\operatorname{\mathcal{C}}'$, which can be described concretely as follows:
The objects of $\operatorname{\mathcal{C}}'$ are the objects of $\operatorname{\mathcal{C}}$: that is, we have $\operatorname{Ob}(\operatorname{\mathcal{C}}') = \operatorname{Ob}(\operatorname{\mathcal{C}})$.
For every pair of objects $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}')$, we set $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}( X,Y) = F( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )$.
For every triple of objects $X,Y,Z \in \operatorname{Ob}(\operatorname{\mathcal{C}}')$, the composition law $c'_{Z,Y,X}$ for $\operatorname{\mathcal{C}}'$ is given by the composition
\begin{eqnarray*} \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y) & = & F( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) ) \otimes F( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \\ & \xrightarrow {\mu } & F( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \otimes \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \\ & \xrightarrow { F(c_{Z,Y,X} ) } & F( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) ) \\ & = & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}( X, Z). \end{eqnarray*}For every object $X \in \operatorname{Ob}(\operatorname{\mathcal{C}}')$, the identity morphism $e'_{X}$ for $X$ in $\operatorname{\mathcal{C}}'$ is given by the composition
\[ \mathbf{1}_{\operatorname{\mathcal{A}}'} \xrightarrow { \epsilon } F( \mathbf{1}_{\operatorname{\mathcal{A}}} ) \xrightarrow { F( e_ X) } F( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) ) = \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,X). \]