II.1.10

This problem has Hartshorne Height 1.

Outline
For every $$U$$ you have the desired factorization on the level of presheaves $$F_i(U) \to \varinjlim F_i(U) \to G(U)$$ $$F_i(U) \to F_j(V) \to \varinjlim F_i(V) \to G(V)$$ is the same as $$F_i(U) \to \varinjlim F_i(U) \to G(U) \to G(V)$$ (use the universal property of direct limit applied to the map $$F_i(U) \to G(V)$$.
 * Fix a sheaf $$G$$ and a system of compatible morphisms $$F_i \to G$$.
 * Show this is compatible with the restriction morphisms, i.e. for $$V\subset U$$ and any F_i \to F_j in the directed system (possibly the identity) then