To be used with questions about continuous homomorphisms into a space carrying both a topological and an algebraic structure.
Let $G$ and $H$ be topological groups. A continuous homomorphism from $G$ to $H$ is a map $\varphi : G \to H$ such that $\varphi$ is both continuous and a homomorphism. These maps are the natural maps between topological groups; they are the morphisms in the category of topological groups.