If one has a topological space $X$ and three presheaves resp. sheaves $F$ and $G$ and $H$ of abelian groups on it with morphisms of presheaves resp. sheaves $F\rightarrow H$, $G \rightarrow H$, then I wonder if one can consider a fiber product of $F$ and $G$ over $H$ in the category of presheaves and sheaves $F \times_HG$
Well, one would perhaps define it in the category of presheaves just as
$F\times _HG (U)$:= all (s,t) $\in F(U)\times G(U)$ going to the same element in $H(U)$.
And for sheaves then take the associated sheaf of this. Or perhaps it is already a sheaf if $F,G,H$ are sheaves?
Just give me some comment if this makes sense and if this concept is of relevance anywhere in Algebraic Geometry. I have never seen it indeed except for perhaps, if you want so, in the case of Schemes, where you consider a scheme as Zariski-Hom-sheaf.