Suppose now that the objects in question are abelian topological groups $G$ so that morphisms are continuous group homomorphisms. Given an exact sequence of abelian topological groups $0 \to G''\to G \to G''$ is the functor $\textrm{Hom}(H,-)$ a left-exact functor? $H$ is an abelian topological group.
We know that if we only care about group homomorphisms then $\textrm{Hom}(H,-)$ is left-exact, but will the condition of continuity now change anything? It seems there is a problem because to prove exactness at $\textrm{Hom}(H,G)$ requires defining a map from $H \to G''$ which may not be continuous.
Edit: PinkElephants has shown us that this functor may fail to be exact. However if we consider a specific case which is the exact sequence
$0 \to \Bbb{Z} \to \Bbb{R} \stackrel{e^{2\pi i x}}{\longrightarrow } S^1 \to 0$
then is the functor $\textrm{Hom}(S^1, -)$ left-exact? What about the contravariant functor $\textrm{Hom}(-,S^1)$, is it still right-exact?