\begin{definition}[Definition:Direct Image of Presheaf/Definition 2]
Let $\mathbf C$ be a category.
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f : T_1 \to T_2$ be continuous.
Let $\FF$ be a $\mathbf C$-valued presheaf on $T_1$.
The '''direct image''' of $\FF$ via $f$ is the $\mathbf C$-valued presheaf $f_*\FF$ that is the composition $\FF \circ \map {\operatorname{Open} } f$, where $\operatorname{Open}$ is the open subsets functor.
\end{definition}