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In Sheaves in Geometry and Logic, Moerdijk and Mac Lane construct the associated sheaf on a site $(\mathcal C, J)$ of a presheaf $P$ as $$ a(P) = (P^+)^+ ,$$ where $P^+$ is defined pointwise as $P^+(U) = \underrightarrow{\mathrm{lim}}_{R \in J(U)} \mathrm{Hom}_{\mathrm{Psh}(\mathcal C)}(R, P)$. Then they prove that $P^+$ is a presheaf (satisfying some properties).

I wonder if one could skip the last part (proving that $P^+$ is a presheaf) by defining $P^+$ direclty as a colimit in the category $\mathrm{Psh}(\mathcal C)$ rather thant pointwise. I tried to find out that colimit without success, principally because the pointwise compuation makes the indexing family dependant of the point. So is it possible ? If so, could someone give me a hint ? If not, what is the reason ?

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Colimits in the category of presheaves are computed pointwise. – Martin Brandenburg Apr 30 '13 at 10:24
@MartinBrandenburg Yes, but with a fixed indexing category, right ? Namely, for a diagram $A : I \to \mathrm{Psh}(\mathcal C)$, one have for $U$ object of $\mathcal C$ $(\underrightarrow{\mathrm{lim}}_I A)(U) = \underrightarrow{\mathrm{lim}}_I A_U$ where $A_U$ is the diagram $A_U(i) = A(i)(U)$ for $i$ object of $I$. I'm trying here to find the $I$ and $A$ for $P^+$. – Pece Apr 30 '13 at 10:29
No, I don't think there's any shortcut here, precisely because the indexing category is different from one place to the next. By formal nonsense there is a diagram whose colimit computes $P^+$ for a given $P$, but it's not obvious to me whether the diagram depends on $P$ or not. – Zhen Lin Apr 30 '13 at 12:26
@ZhenLin Thanks, that was precisely my question (the dependence on $P$ of such a diagram). The existence of the diagram by itself does not interest me per se. I would have been happy to find an expression of the diagram with respect to $P$. – Pece Apr 30 '13 at 12:35

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