Yoneda lemma for enriched categories

Question

Let $\mathcal{M}$ be a monoidal category. Is there a generalization of the Yoneda lemma to categories enriched over $\mathcal{M}$?

In the specific case I need, $\mathcal{M}$ would be the category $k\operatorname{-Vect}$ of $k$-vector spaces.

Answer 1

Yes, see here. A standard reference is Kelly's Basic concepts of enriched category theory, TAC reprint. The case of $k$-linear categories is very simple (basically because here the monoidal unit is a generator). Here, $k$ can be any commutative ring. For $k$-linear functors $F : \mathcal{C} \to \mathsf{Mod}_k$ and objects $A \in \mathrm{Ob}(\mathcal{C})$ there is a canonical $k$-linear map $$\mathrm{Hom}\bigl(\mathrm{Hom}(A,-),F\bigr) \to F(A)$$ which is an isomorphism since by the usual Yoneda Lemma the underlying map of sets is an isomorphism.