Let's say $X$ is a topological space and let's consider the categories of sheaves and presheaves of abelian groups on $X$. Suppose we have a presheaf that is a direct summand (in the category of presheaves) of a sheaf. Can we conclude that the presheaf is also a sheaf? What if we assume further that the complementary direct summand is a sheaf too--does that help us?
I know that in general colimits in the category of sheaves differs than those in the category of presheaves, and in general the quotient of two sheaves (in the category of presheaves) need not be a sheaf (I think). But maybe the fact that things are split helps us?
What about in the general context of sheaves and presheaves on a site?