Infinite Galois equivalence 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!
 A: When $X$ is connected, we see that any graph $\Gamma_f$ for $f \colon X \to Y$ is connected as well, so we get a bijection
\begin{align*}
\mathscr C(X,Y) &\to \left\{\Gamma \subseteq X \times Y \text{ connected}\ \bigg|\ \Gamma \underset{\pi_1}{\overset{\sim}\to} X \right\} \\
f &\mapsto \Gamma_f.
\end{align*}
The same holds in $G\text{-}\mathbf{Set}$ for any Noohi group $G$. Writing $G = \pi_1(\mathscr C,F)$ for simplicity, we see that $\mathscr C(X,Y) \to \operatorname{Hom}_G(FX,FY)$ is a bijection for $X$ connected because $F \colon \mathscr C \to G\text{-}\mathbf{Set}$ preserves connected components (and products, etc).
Now I suppose the general result follows from axiom (2) or (3) of infinite Galois categories (even though it's not completely spelled out what (3) means). But at the very least we know that any object $X$ is a coproduct $\coprod_{i \in I} X_i$ of connected objects $X_i$. Since $F$ preserves colimits, we get
$$\begin{array}{ccccc}
\mathscr C(X,Y) & \cong & \mathscr C\Big(\coprod\limits_{i \in I} X_i,Y\Big) & \cong & \prod\limits_{i \in I} \mathscr C\big(X_i,Y\big)\\ & & & & \downarrow\wr\!\! \\ \operatorname{Hom}_G(FX,FY) & \cong & \operatorname{Hom}_G\bigg(\coprod\limits_{i \in I} FX_i,FY\bigg) & \cong & \prod\limits_{i \in I} \operatorname{Hom}_G\big(FX_i,FY\big)
\end{array}$$
by the connected case presented above. $\square$
This type of argument might be considered 'standard' when dealing with Galois categories (see for instance [Tag 0BN0 (7)]), which would explain why no details were given.
