Let $X$ be a topological space and $\{V_i\}$ a cover of $X$. Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be a family of sheaves. One can glue this family to obtain a sheaf $F:\mathsf{Open}(X)^{\mathrm{op}}\to \mathsf{Sets}$, iff the family satisfies some local compatibility conditions like the cocycle condition on threefold unions. This is the situation for sheaves on a topological space.
Is there a similar statement for sheaves on a Grothendieck site $C$, in particular for the site $\mathsf{Sch}$ of schemes with the Zariski- or the etale topology?
Perhaps there is a sheaf $F:C^{\mathrm{op}}\to \mathsf{Sets}$ iff there is a compatible family $F:C^{\mathrm{op}}/V_i\to \mathsf{Sets}$ of sheaves on the slice topoi? What are the compatibility conditions then? Does somebody have a reference?
My second question is most probably nonsense but it would be nice if somebody could support this such that I can forget about this idea forever, even for the situation of sheaves on a topological space: Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ and $G_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be two families of sheaves and $F_i\to G_i$ a morphism of sheaves. Does this glue to a morphism $F\to G$ of the glued sheaves?
Yes, a reference is Stacks Project 7.23. The proofs there are quite sketchy. The reason is that they work exactly as for topological spaces. Just replace open subsets by objects of the site, open coverings by coverings in the abstract sense, intersections by fiber products, sections by morphisms, subspace topology by slice topology.