Let $X$ be a topological space and let $K$ be a field.
If $F$ is a sheaf of $K$-vector spaces over $X$, can $F$ be produced from constant sheaves $K_{U_i}$ on some open subsets$U_i \subset X$ by taking direct limit?
In other words, is there any directed system $(K_{U_i}, f_{ij})$ that satisfies $F = \varinjlim K_{U_i}$
If it is true, how can I prove it?