If $f:X\to Y$ is a continuous map of topological spaces, and if ${\bf Sheaves}(X)$ is the category of sheaves of abelian groups on $X$ (and similarly for ${\bf Sheaves}(Y)$), then the direct image functor $f_{*}:{\bf Sheaves}(X)\to{\bf Sheaves}(Y)$ sends a sheaf $\mathcal{F}$ on $X$ to its direct image $f_{*}\mathcal{F}$ on $Y$. A morphism of sheaves $g:\mathcal{F}\to\mathcal{G}$ obviously gives rise to a morphism of sheaves $f_{*}g:f_{*}\mathcal{F}\to f_{*}\mathcal{G}$, and this determines a functor.

If $\mathcal{F}$ is a sheaf of abelian groups (or anything else), so is $f_{*}\mathcal{F}$, so likewise we get direct image functors $f_{*}:{\bf Ab}(X)\to{\bf Ab}(Y)$, where ${\bf Ab}(X)$ is the category of sheaves of abelian groups on $X$.