When can functors fail to be adjoints if their hom sets are bijective? Suppose we have categories $\mathcal{C}$ and $\mathcal{D}$ with some appropriate adjectives like local smallness along with a pair of functors $F: \mathcal{C} \rightarrow \mathcal{D}$ and $G: \mathcal{D} \rightarrow \mathcal{C}$. Suppose further that for every pair of objects $A$ in $Ob(\mathcal{C})$ and $B \in Ob(\mathcal{D})$ we have a bijection of sets
$$
\text{Hom}_{\mathcal{D}}(F(A), B) \simeq \text{Hom}_{\mathcal{C}}(A, G(B)), 
$$
but say we do not assume a priori that this is an isomorphism of bifunctors, that is, presumably naturality in $A$ or $B$ may fail. Is this possible? I strongly suspect it is, because naturality seems like an essential part of an adjunction, but are there explicit counter examples? Or will all such counter examples be fairly artificial constructions like a finite category, etc? 
As an extension, suppose we knew (via some adjoint functor theorem) that (WLOG) $F$ did indeed have a right adjoint. Would the above bijection be enough to conclude that it must be $G$, or can there be non-isomophic functors that agree with an adjoint pair on objects only?
Similarly, if we were able to show naturality in (WLOG) $A$, and we knew an adjoint existed, is there some Yoneda argument that can be made to force naturality in $B$?
I'm sure these may not be too difficult to work out for someone with experience, but I'm still relatively new to adjoint functors so I'm not sure I'd trust my own answer even if I did manage to figure anything out.

For the first question, let $F$ be the duality functor on finite-dimensional vector spaces and $G$ the identity functor. One knows $F(V)\cong V$ non-naturally, so $\mathbf{Vect}(F(V),W)\cong \mathbf{Vect}(V,W)$, non-naturally. I assume this counts as a natural example, since it's the standard example of an unnatural isomorphism. Derek answered the last questions in the comments.
