Definition:Direct Image of Presheaf/Definition 2

Definition
Let $\mathbf C$ be a category.

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $f : T_1 \to T_2$ be continuous.

Let $\mathcal F$ be a $\mathbf C$-valued presheaf on $T_1$.

The direct image of $\mathcal F$ via $f$ is the $\mathbf C$-valued presheaf $f_*\mathcal F$ that is the composition $\mathcal F \circ \operatorname{Open} \left({f}\right)$, where $\operatorname{Open}$ is the open subsets functor.

Also see

 * Equivalence of Definitions of Direct Image of Presheaf