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 ?