Compatibility of pullbacks with an equivalence relation

Question

This question was originally posted last week in Math Stack Exchange (see here).

I'm currently working on the proof of the existence of the sheafification in Angelo Vistoli’s 2007 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: $$ a,b\in F(U):a\sim b\Rightarrow f^*(a)\sim f^*(b) $$

Question: How do I prove this?

Answer 1

After I saw my mistake I got the proof really quick. We need 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, $pr_1^*(f^*(a))=(pr_1^*\circ f^*)(a)=pr_2^*(\phi^*(a))=pr_2^*(\phi^*(b))=(pr_1^*\circ f^*)(b)=pr_1^*(f^*(b)).$

Since $pr_1$ is a covering (Covering Axioms), we get $f^*(a)\sim f^*(b)$.

q.e.d.