I'm currently working on the proof of the existence of the sheafification in Notes on Grothendieck topologies, fibered categories and descent theory , but i currently stuck on a statement in the proof of Theorem 2.64. $(ii)$. That is the following:
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 $\phi^*(a)=\phi^*(b)$.
Now the statement is, that for every morphism $f:S\rightarrow U$ the pullback $F(f)=f^*:F(U)\rightarrow F(S)$ is compatible with $\sim$. That means:
$$
a,b\in F(U):a\sim b\Rightarrow f^*(a)=f^*(b)
$$
I tried proving this statement with the fibre product of $f$ and $\phi$, where $f$ is an arbitrarily morphism and $\phi$ the covering from $a\sim b$.
So we get the following commutative diagram:
$\require{AMScd}$
\begin{CD}
S\times_UT @>{pr_2}>> T\\
@V{pr_1}VV @VV{\phi}V\\
S @>{f}>> U
\end{CD}
hence $$ \phi\circ pr_2=f\circ pr_1. $$ Using $F$ on both sides we get $$ F(\phi\circ pr_2)=F(f\circ pr_1)\quad\text{ or }\quad pr_2^*\circ\phi^*=pr_1^*\circ f^*. $$ Hence, $pr_1^*\circ f^*(a)=pr_2^*\circ\phi^*(a)=pr_2^*\circ\phi^*(b)=pr_1^*\circ f^*(b).$
Now the questions:
- Is $pr_1^*$ monic or is it already proven?
- Is there a different way to proof this? If yes, i just need a starting point.