Plus construction of sheafification as a colimit of presheaves. 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 ?
 A: There is a shortcut via Kan extensions.  I learned it from section 6.2.2 of Higher Topos Theory, which treats the case of sheaves of simplicial sets on a general site.  Here's what it looks like for sheaves of sets on a topological space.
Let $\mathcal{C}$ be the frame of open subsets of some topological space $X$.  Let $\text{Cov}(\mathcal{C})$ be the poset in which an object is a pair $(U, \mathfrak{U})$ of an object $U$ of $\mathcal{C}$ and an open cover $\mathfrak{U}$ of $U$, and $(U, \mathfrak{U}) \leq (V, \mathfrak{V})$ if $U \subseteq V$ and $\mathfrak{U}$ is a refinement of $\mathfrak{V}$.  Let $\mathcal{C}^{+}$ be the poset of triples $(U, \mathfrak{U}, U')$ such that $(U, \mathfrak{U})$ is an object of $\text{Cov}(\mathcal{C})$ and $U'$ is a subset of some element of $\mathfrak{U}$, and $(U, \mathfrak{U}, U') \leq (V, \mathfrak{V}, V')$ just in case $(U, \mathfrak{U}) \leq (V, \mathfrak{V})$ in $\text{Cov}(\mathcal{C})$ and $U' \subseteq V'$.
There are three projection functors
$$
e : \mathcal{C}^{+} \to \mathcal{C} \qquad \pi : \mathcal{C}^{+} \to \text{Cov}(\mathcal{C}) \qquad \rho : \text{Cov}(\mathcal{C}) \to \mathcal{C}
$$
and the plus construction is the composite
$$
[\mathcal{C}^{\text{op}}, \mathsf{Set}] \xrightarrow{e^{*}}
[(\mathcal{C}^{+})^{\text{op}}, \mathsf{Set}] \xrightarrow{\pi_{*}}
[\text{Cov}(\mathcal{C})^{\text{op}}, \mathsf{Set}] \xrightarrow{\rho_{!}}
[\mathcal{C}^{\text{op}}, \mathsf{Set}]
$$
where the functor $e^{*}$ is pullback along $e$, the functor $\pi_{*}$ is the right adjoint of the pullback $\pi^{*}$, and the functor $\rho_{!}$ is the left adjoint of the pullback $\rho^{*}$.  So $P^{+}$ is a presheaf because the plus construction is a functor, giving the shortcut you want.  
These adjoints are given by pointwise Kan extensions, so for any $U$ we can compute $P^{+}(U)$ using (co)limits.  Some finality arguments show that we recover the definition you give of $P^{+}(U)$.
