Sheaf commutes with limits

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 $M$.

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1 Answer

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.

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It's quite irritating that this myth keeps showing up. I wonder where it comes from? – Zhen Lin Jan 9 '14 at 17:35