TITLE: Adjoint functors requiring a natural bijection
QUESTION [13 upvotes]: 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?

REPLY [3 votes]: Given a natural transformation 
$$\varphi_{} \colon \mathbf A(-,G(-)) \to \mathbf B(F(-),-) $$ 
this is the same as a family natural transformation between the functors
$$\varphi_Y \colon \mathbf A(-,G(Y)) \to \mathbf B(F(-),Y)$$
natural in $Y$.
For every $Y \in \mathbf B$ by yoneda lemma we have that, if $\epsilon_Y=\varphi_Y(1_{G(Y)})$, then 
$$\varphi_Y(f)=\mathbf B(F(f),Y)(\epsilon_Y)=\epsilon_Y \circ F(f)$$
for every $f \in \mathbf A(X,G(Y))$.
The requirement that $\varphi$ is an isomorphism implies that $\varphi_Y$ are all isomorphisms and so $\epsilon_Y$ must be universal (and that's why the adjoint are important, because they make arise universal objects).
If you drop the naturality condition you cannot use yoneda and so you cannot get the universal morphism.
That's why we need naturality. :)
Hope this helps.