Adjoint functors requiring a natural bijection

Question

When showing that two functors $F:A\rightarrow B$ and $G:B\rightarrow A$ are adjoint, one defines a natural bijection $\mathrm{Mor}(X,G(Y)) \rightarrow \mathrm{Mor}(F(X),Y)$. What if one do not require the bijection to be natural, what issues would arise?

Answer 1

Adjunction of $F,G$ is a bridge between $\mathcal A$ and $\mathcal B$, in most of the examples so called heteromorphisms are definable from objects of $\mathcal A$ to that of $\mathcal B$, and these have to (naturally!) correspond to elements of both homsets $\mathrm{Mor}(FX,Y)$ and $\mathrm{Mor}(X,GY)$. Then $F$ can be obtained by reflections and $G$ by 'coreflections' in this bigger category which disjointly contains $\mathcal A$ and $\mathcal B$ and the heteromorphisms defined by the adjunction. Naturality is crucial.