Yoneda lemma for representable profunctors?

Question

For $F:C\rightarrow D$ a functor, consider the profunctor $h_F:=\operatorname{Hom}_D(-,F-):D^{op}\times C \rightarrow \mathsf{Set}$. For a functor $G:C\rightarrow D$, define $h_G$ similarly. Is it true that $\operatorname{Nat}(h_F,h_G)\cong \operatorname{Nat}(F,G)$ holds? If so, does this bijection restrict to a bijection between the set of natural isomorphisms? I suppose the former is equivalent to the question: Is the functor $[C,D]\rightarrow [C\times D^{op},\mathsf{Set}];F\mapsto h_F$ fully faithful? This functor is (under currying) precisely postcomposition with the fully faithful Yoneda functor $Y:D\rightarrow [D^{op},\mathsf{Set}]$.

Answer 1

In slightly more generality, given a category $\mathcal{C}$ and a fully faithful functor $i\colon\mathcal{D}\hookrightarrow\mathcal{E}$, it holds that postcomposition with $i$ gives a fully faithful functor $i_*\colon\mathrm{Fun}(\mathcal{C},\mathcal{D})\hookrightarrow\mathrm{Fun}(\mathcal{C},\mathcal{E})$. If you think about the data of which a natural transformation consists, this is immediately clear: for two functors $F,G\colon \mathcal{C}\to\mathcal{D}$, a natural transformation $\eta\colon i\circ F\to i\circ G$ consists of maps $\eta_c\colon iF(c)\to iG(c)$ for all $c\in\mathcal{C}$, subject to some relations. But since $i$ is fully faithful, there is a unique way to see $\eta_c$ as a morphism in $\mathcal{D}$, and this gives us a uniquely determined natural transformation $F\to G$.

Answer 2

Yes. If $c \in C$ is fixed, a family of natural morphisms $\hom(d,F(c)) \to \hom(d,G(c))$ for $d \in D$ is the same as a morphism $F(c) \to G(c)$ by the Yoneda Lemma. Now, the morphisms $\hom(d,F(c)) \to \hom(d,G(c))$ are natural in $c$ iff the morphisms $F(c) \to G(c)$ are natural in $c$. Thus, we have a morphism $F \to G$.