Consider a monoidal category $\mathcal{C}$ which is enriched over $\mathrm{vect_k}$ (=finite dimensional vector spaces over a field $k$). $\mathcal{C}$ is abelian and semi simple. If it helps one can assume that $\mathcal{C}$ is rigid and has finitely many irreducble objects and irreducible unit object.
I want to show that the functor $$F_{D, V}: \mathcal{C} \rightarrow \mathrm{vect_k}, C \mapsto V \otimes_k \mathrm{Hom}(C, D)$$ is representable.
On ordinary algebra basis I found that there is a connection between semi simple and projective and I think I heard something that a functor is representable if every object is projective.
But I guess that's only random, messy information clustered. Does anyone have more detailed information? A source to an article or book?
Thanks in advance for any help!