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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 $i\circ F \cong i\circ Hom (B,-)$ ?

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  • $\begingroup$ 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? $\endgroup$ – Qiaochu Yuan Sep 9 '19 at 23:01

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