Given a presheaf $F$, in Notes on Grothendieck topologies, fibered categories and descent theory there is a construction of the sheafification (Proof for theorem 2.64).
The first part is the construction of a separared functor $F^S$ and natural transformation $F\rightarrow F^S$:
Note: In this context, a Grothendieck topology is a Singleton Grothendieck topology, such that every covering of an object $U$ in $C$ is a single map $\phi:T\rightarrow U$.
We have a site $C$, that is a category $C$ with a Grothendieck topology. Then we have a functor $$F:C^{op}\rightarrow Set$$ We define an equivalence relation $\sim$ for every object $U$ of $C$ on $F(U)$ as: $a\sim b$ if there exists a covering $\phi:T\rightarrow U$, such that the pullback $F(\phi)=\phi^*$ of $a$ and $b$ coincide in $F(T)$. In other words $\phi^*(a)=\phi^*(b)$. Then define $F^S(U):=F(U)/\sim$. Now, that for every morphism $f:S\rightarrow U$ the pullback $F(f)=f^*:F(U)\rightarrow F(S)$ is compatible with $\sim$. That means: $$ a,b\in F(U):a\sim b\Rightarrow f^*(a)=f^*(b). $$ That yields us the pullback $F^S(f):F^S(U)\rightarrow F^S(V)$ and we get the functor $F^S$ and a natural transformation $(S):F\Rightarrow F^S$.
The second part is quite analogue. We have a given separated presheaf $F'$ (I could use $F$, but I want to avoid some notational confusion). Then we define for an object $U\in C$ the set $P_U:=\{(\phi,f)\,|\,\phi:T\rightarrow U \text{ is a covering, }f\in F'(T)\text{ such that }pr_1^*f=pr_2^*\text{ in }F'(T\times_UT)\}$. On this set we impose an equivalence relation, by declaring $(\phi,f)\sim'(\phi',f')$ if $pr_1^*f=pr_2^*f'$ in $F'(T\times_UT')$. Now we define $F^+(U):=P_U/\sim'$. Given an arrow $X\rightarrow U$ in $C$, we define the a function $F^+(U)\rightarrow F^+(X)$ by $[\phi:T\rightarrow U,f]\mapsto [pr_1:X\times_UT\rightarrow X,pr_2^*f]$. This is from the fiber product $\require{AMScd}$ \begin{CD} X\times_UT @>{pr_2}>> T\\ @V{pr_1}VV @VV{\phi}V\\ X @>>> U. \end{CD} That defines our functor $F^+$. There is also a natural transformation $(+):F'\Rightarrow F^+$, obtained by sending an element $f\in F'(U)$ into $(U\rightarrow U, f)\in F^+(U)$.
I proved, that $F^S$ is a separated presheaf and $F^+$ a sheaf. But I don't know how to prove the following statements:
- Is $\Phi:F\Rightarrow G$ a natural transformation from a presheaf $F$ to a separated presheaf $G$, than it factors uniquely through $F^S$.
- The composite of the natural transformations $F\Rightarrow F^S$ and $F^S\Rightarrow F^+$ has the desired universal property. (Resp. Is $\Phi:F'\Rightarrow G'$ a natural transformation from a separated presheaf $F'$ to a sheaf $G'$, than it factors uniquely through $F^+$.)
I found a similar question Universal property of sheafification, but they are using a different definition (Hartshorne).
To prove these statements I have to contruct natural transformations $\Psi:F^S\Rightarrow G$ for 1. and $\Psi':F^+\Rightarrow G'$ for 2.
Update: For $\Psi:F^S\Rightarrow G$ we have the components $\Psi_U:=\iota\circ \psi_U$, for all $U\in C$ with $\iota:\Phi(F(U))\hookrightarrow G(U)\,|\,\Phi_U(x)\mapsto\Phi_U(x)$ and $\psi_U:F^S(U)[=F(U)/\sim]\rightarrow \Phi(F(U))\,|\,[x]\mapsto\Phi_U(x)$. Hence, with $\pi_U:F(U)\rightarrow F^S(U)\,|\,x\mapsto [x]$, we get the commutative diagram $\require{AMScd}$ \begin{CD} F(U) @>{\Phi_U}>> G(U)\\ @V{\pi_U}VV @VV{Id_{G(U)}}V\\ F^S(U) @>{\iota\circ\psi_U}>> G(U). \end{CD}
But I don't know how to contruct and $\Psi'$.