There is such statement:
Presheaf $\mathfrak F$ is a sheaf if it commutes with limits: $$\mathfrak F(\lim_{\to} U_\alpha)=\lim_{\gets}\mathfrak F(U_\alpha).$$
Can you rewrite this formula more exactly using following definitions.
Let $\Phi : \mathfrak J \to \mathfrak C$ is a functor between two categories.
Definition 1. Direct limit of $\Phi$ is an object $X\in\mathfrak C$ which represent functor $${\mathfrak C}^0\longrightarrow Sets$$ $$Y\longmapsto\hom_{Funct(\mathfrak J,\mathfrak C)}(\Delta Y, \Phi),$$ where $\Delta : \mathfrak C\to Funct(\mathfrak J,\mathfrak C)$ be a diagonal functor.
Definition 2. Analogically inverse limit corepresent functor $$\mathfrak C\longrightarrow Sets$$ $$Y\longmapsto\hom_{Funct(\mathfrak J,\mathfrak C)}(\Phi,\Delta Y).$$
Definition 3. At last presheaf on space $M$ is a functor from category of open subsets of $M$.
The statement is false. But it is correct that a sheaf is a presheaf which preserves a certain class of colimits. See also my answer here.