This is a rather open-ended question, but the title basically says it all.
Given a topological space $X$, and a target category $\mathfrak{C}$, there is a notion of a sheaf over $X$. This is an abelian category under the assumption that $\mathfrak{C}$ is. Suppose we talk about nice topological spaces, such as manifolds or varieties, and nice target categories, such as abelian groups, modules, vector spaces, or whatever other nicety you prefer.
Is the category of sheaves over some object of your nice topological category a small category in general?