## FANDOM

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This problem has Hartshorne Height 1.

### Sketch

• Fix a sheaf $G$ and a system of compatible morphisms $F_i \to G$.

For every $U$ you have the desired factorization on the level of presheaves $F_i(U) \to \varinjlim F_i(U) \to G(U)$

• 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
$F_i(U) \to F_j(V) \to \varinjlim F_i(V) \to G(V)$

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 $F_i(U) \to G(V)$).

• Use Proposition-Definition II.1.2