Is the Alexandroff one-point compactification of a locally compact Hausdorff space ($\mathbf{LCHaus}$) a functor to the category of compact Hausdorff spaces ($\mathbf{CHaus}$)? It seems to me that one has to consider only proper continuous maps as morphisms.
If one does so, then the most natural definition for the induced map $\hat f\colon\hat X\to \hat Y$ seems to work (just send $x\mapsto f(x)$ and $\infty_X\mapsto \infty_Y$, continuity has to be checked only on open sets containing $\infty$) but a lot of interesting situations seem to be excluded: one hopes that (e.g.) a map $h\colon (0,1)\to \mathbb R$ such that $\lim_{x\to 1^-}h(x)=\lim_{x\to 0^+}h(x)$ admits an extension $\hat h\colon \widehat{(0,1)}\to \mathbb R$; but what if $h$ is not proper? It is "outside" the category I'm considering. So:
How can one define suitable topological categories between which $\widehat{(-)}$ is a functor?
Morally a compactification should be a left adjoint to an inclusion (in this case $\iota\colon \mathbf{CHaus}\hookrightarrow\mathbf{LCHaus}$), but even if it seems evident that (taking only proper maps) $\hom_{\mathbf{CHaus}}(\hat X,Y)\cong \hom_{\mathbf{LCHaus}}(X,\iota Y)$ the Alexandroff correspondence seems to be ill-behaved with respect to colimits... is there any hope to make $X\mapsto \hat X$ adjoint to something?
Thanks for your attention