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I'm trying to prove this: Let $(\mathcal{C}, J)$ be a site and suppose that for every $X\in ob(\mathcal{C})$ and $R\in J(X)$ the family $\{ \bar{f}_Y: \mathcal{Y}_Y\rightarrow \mathcal{Y}_X \}_{Y\in ob(\mathcal{C}), f\in R(Y)}$ is an epimorphic family then the sub-sheaf $\mathcal{Y}_X=Hom(-,X)$ is a sheaf in Let $(\mathcal{C}, J)$. My strategy is to prove that for every $Y\in ob(\mathcal{C})$ and $S\in J(X)$ we have $Hom(\mathcal{Y}_Y,\mathcal{Y}_X)\cong Hom(S,\mathcal{Y}_X)$ The point is that I don't know how to involved the given epimorphic family $\{ \bar{f}_Y: \mathcal{Y}_Y\rightarrow \mathcal{Y}_X \}_{Y\in ob(\mathcal{C}), f\in R(Y)}$, in order to show that isomorphism. Any hint? Thanks.

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Jointly epimorphic in which category? Presheaves? Sheaves for the canonical topology? Be more precise! –  Zhen Lin Jun 5 at 11:46
    
Yes sorry, jointly epimorphic family in the category of presheaves, and we want $\mathcal{Y}_X$ to be a sheaf for the canonical topology. –  user155330 Jun 5 at 15:01
    
Actually, your question doesn't make any sense at all. $\mathrm{Hom}(-, X)$ is always a sheaf for the canonical topology, by definition. Your isomorphism involving $S$ does not type-check either. –  Zhen Lin Jun 5 at 15:07
    
$\mathcal{Y}_X$ must be sheave on $(\mathcal{C},J)$ –  user155330 Jun 5 at 15:23
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Well, if $R$ really results in a jointly epimorphic family in the category of presheaves, then $R$ must be the maximal sieve. So your topology is "trivial" and every presheaf is a sheaf. –  Zhen Lin Jun 5 at 15:43

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