What are the epimorphisms in the category of topological groups?

Question

A morphism $f: X \to Y$ is an epimorphism if for all $g, h: Y \to Z$, if $g \circ f = h \circ f$ then $g = h$. The epimorphisms in the category of groups are the surjective group homomorphisms. The epimorphisms in the category of topological spaces are the continuous surjections. Are the epimorphisms in the category of topological groups the continuous surjective homomorphisms?

Answer 1

A cool answer is this one. Sorry if this the same proof given in the reference in the commentaries.

Consider the forgetful functor

$$U: TopGrp \rightarrow Grp$$

For any group $G$ let $L(G)$ be the the same group with the discrete topology and $R(G)$ be with the codiscrete topology. Then

$$TopGrp(L(G),H)=Grp(G,U(H))$$

since every function $L(G)\rightarrow H$ is continuous. We also have that

$$TopGrp(G,R(H))=Grp(U(G),H)$$

since every function $G \rightarrow R(H)$ is continuous.

Thus we conclude that $U$ is both right and left adjoint. Right adjoints preserve monomorphisms and left adjpints preserve epimorphisms. We conclude that a morphism of topological groups is an (mono)epimorphism if, and only if, its underlying morphism of groups is.