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> Let $S$ be a discrete topological space. Let $p:X \times S \rightarrow X$ be a projection. $F$ is a sheaf on $X\times S$ and let $F\_s=i\_s^\*(F),$ where $i\_s(y)=(y,s) $. Show that $p\_\*(F) \simeq \prod\_{s \in S} F\_s.$
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In first sight this statement looks easy, but when I tried to make this exercise, I don't have idea how associate $F(p^{-1}(U))$ with product and $colim$ at right side (I think, here $colim$ commute with product because we work under filtered category, I'm right?).
Could anyone give me any hint how to do this (I think easy) exercise? I would be grateful.