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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?

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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.2010-08-16
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    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.2010-08-16
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    More generally, one could ask: Why is a natural isomorphism between two functors actually required to be natural?2010-08-16

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