I am having some trouble trying to understand the proof of Theorem 7.2.5 in Bhatt and Scholze's paper The pro-étale topology for schemes. Specifically, I don't quite understand why it was necessary to prove that $F: C \to \mathit{Sets}$ preserves connectedness of objects, and how viewing morphisms $f: X \to Y$ in $C$ as monomorphisms $\Gamma_f: X \to X \times Y$ can help us prove that $F: C \to \pi_1(C, F)\text-\mathit{Sets}$ is fully faithful (I could only show that $F: C \to \mathit{Sets}$ is fully faithful).

Any help is very much appreciated!