This problem has Hartshorne Height 1.
- Fix a sheaf and a system of compatible morphisms .
For every you have the desired factorization on the level of presheaves
- Show this is compatible with the restriction morphisms, i.e. for and any in the directed system (possibly the identity) then
is the same as
(use the universal property of direct limit applied to the map ).
- Use Proposition-Definition II.1.2