# Adjoint functors requiring a natural bijection

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

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Horrible, horrible issues would arise. It is very easy to find non-natural bijections, so I'm not sure what you mean. –  Qiaochu Yuan Aug 16 '10 at 19:27
This is more or less asking what if happens we do not require of a map between two groups to preserve the group operations. The answer is: you would end up with a fairly useless concept. –  G. Rodrigues Aug 16 '10 at 19:32
More generally, one could ask: Why is a natural isomorphism between two functors actually required to be natural? –  Rasmus Aug 16 '10 at 20:37
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 $Mor(FX,Y)$ and $Mor(X,GY)$. 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.