It is a well known fact that in CGWH spaces we have following homeomorphism:
$$\mathrm{map}(X \times Y,Z)\cong\mathrm{map}(X, \mathrm{map}(Y,Z)).$$
The proof starts like this:
- We are dealing with locally compact space therefore $\mathrm{ev}_x:X\times \mathrm{map}(X,Y)\to Y$, $\mathrm{inj}_x:X\to \mathrm{map}(Y,Y\times Y)$ (respectively evaluation and injection) are continuous maps. Why is that statement true?
Moreover it is well known that in CGWH category functor $\mathrm{map}(X,\square)$ is right adjoint to $\square \times X$. We have just shown bijection (even homeomorphism) between proper hom-sets but we need to show its naturality, too, am I right? I have seen such reasoning for a few times and each time author didn't bother with proving anything more than bijection of hom-sets. Is that such an obvious thing? I have come up with an idea to use Yoneda lemma once again i.e. we have bijection between $\mathcal{Nat}(\mathrm{map}(X\times Y,\square), \mathrm{map}(X, \mathrm{map}(Y,\square)))$ and $\mathrm{map}(X, \mathrm{map}(Y,X \times Y))$ so if only the last set is non-empty then some natural transformation in $Z$ exists. And similiary we may show naturality in $Y$ by considering $\mathcal{Nat}(\mathrm{map}(\square, \mathrm{map}(X, Z)), \mathrm{map}(X \times \square,Z))$ and claiming that $\mathrm{map}(X \times \mathrm{map}(X, Z),Z)$ is non-empty. Is this reasoning true and shows the lacking part?