Fiber product of sheaves

Solution 1:

Let $X$ be a topological space. The category of sheaves of sets on $X$, $\textrm{Sh}(X)$, is an example of a Grothendieck topos and is, in particular, complete and cocomplete. Therefore fibre products, or pullbacks as they are known in general category theory, exist in $\textrm{Sh}(X)$. Of course, one needs to verify that the forgetful functor $\textbf{Ab}(\textrm{Sh}(X)) \to \textrm{Sh}(X)$ preserves pullbacks, but this is straightforward.

Your proposed construction works and in fact does not require sheafification since limits and colimits of presheaves are constructed sectionwise, and the inclusion of categories $\textrm{Sh}(X) \hookrightarrow \textrm{Psh}(X)$ preserves all limits. (This is because it has a left adjoint, namely the associated sheaf functor.)

The fibre product of schemes is not an example of a fibre product of sheaves, unless you are thinking of a scheme as a sheaf in the Zariski topos.