# Representability of functors in categories other than Set category

Let $$\mathcal C,\mathcal D$$ be locally small categories and also assume that $$\mathcal D$$ is small and that every morphism in $$\mathcal D$$ is a function between sets. Assume that for every $$A,B \in \mathcal Ob (C)$$, we have $$Hom(A,B)\in Ob(\mathcal D)$$ and for every $$A,B,X \in Ob(\mathcal C)$$ and $$\phi \in Hom(A,B)$$, we have $$\hat \phi \in Hom_{\mathcal D} (Hom(X,A), Hom(X,B))$$, where $$\hat \phi (f)=\phi \circ f, \forall f \in Hom (X,A)$$.

So that for every $$X\in Ob(\mathcal C)$$, we have $$Hom(X,-):\mathcal C \to \mathcal D$$ is a co-variant functor.

Let $$i:\mathcal D \to \mathcal Set$$ be the forgetful functor.

Let $$F: \mathcal C \to \mathcal D$$ be a covariant functor .

Then is it true that: $$\exists A \in Ob(\mathcal C)$$ such that $$F \cong Hom(A,-)$$ if and only if $$\exists B\in Ob(\mathcal C)$$ such that $$i\circ F \cong i\circ Hom (B,-)$$ ?

• The setup of the question is a little messy; it sounds like you want $D$ to be a concrete category and you want $C$ to be enriched over $D$. Is that right? If so, then a priori you need to specify an additional piece of information, which is a choice of monoidal structure on $D$. Perhaps you also want the forgetful functor to be lax monoidal or something. What examples do you have in mind? – Qiaochu Yuan Sep 9 '19 at 23:01