Let $LCH$ be the category of locally compact Hausdorff spaces with proper continuous maps.
Question. What are the epimorphisms in $LCH$?
I suspect them to be surjective, but I haven't been able to prove it.
Here is an idea: Let $f : X \to Y$ be an epimorphism. The image $f(X)$ is closed. It follows that $f(X)^+$ is closed in $Y^+$, where $+$ denotes the Alexandrov compactification. If $y \in Y \setminus f(X)$, by Urysohn's Lemma there is some $g \in C(Y^+)$ with $g(y)=1$ and $g(f(X)^+)=0$. Now, $g(\infty)=0$ implies that $g$ restricts to some $g \in C_0(Y)$ such that $g(y)=1$ and $g(f(X))=0$. The remaining problem is that $g$ might be not proper.
I have also tried to prove it in the dual category, which is the category $CommC^*Alg$ of commutative $C^*$-algebras with non-degenerate $*$-homomorphisms. I already know that surjections become injections under this duality, so that the question would be: Is every monomorphism in $CommC^*Alg$ injective? Again, the restrictive morphisms cause some problems in proving this.