enriched functor

Let \(\mathcal{V}\) be a monoidal category and \(\mathcal{C}, \mathcal{D}\) enriched categories over \(\mathcal{V\)} An enriched functor is a functor \(\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}\), such that for \(A,B \in \mathrm{Ob}(\mathcal{C})\)

\begin{align*} \mathcal{F}: \mathrm{Hom}_{\mathcal{C}}(A,B) \rightarrow \mathrm{Hom}_{\mathcal{D}}(\mathcal{F}(A), \mathcal{F}(B)) \end{align*}

is a morphism on hom objects