Functors from connections to a discrete category are never faithful

Let $X$ be a finite directed graph with at least one vertex, and let $G$ be a nontrivial group. Let $\mathcal A_G(X)$ be the groupoid whose objects are $G$-connections $A : E(X)\to G$ and whose arrows $A\to B$ are gauge transformations $g : V(X)\to G$ with $g\cdot A = B$, where $(g\cdot A)(e) = g(\mathrm{src}\,e)\,A(e)\,g(\mathrm{tgt}\,e)^{-1}$.

Statement. For every set $\alpha$ and every functor $F : \mathcal A_G(X)\to \mathrm{Disc}(\alpha)$ into the discrete category on $\alpha$, $F$ is not faithful.

Proof idea. Take $a\in G$ with $a\neq 1$ and the trivial connection $A\equiv 1$. The constant gauge transformation $g\equiv a$ fixes $A$, since $a\cdot 1\cdot a^{-1}=1$, so it is an automorphism of $A$ different from the identity. A discrete category has only identity arrows, so $F(g)=F(\mathrm{id}_A)$.

This is the common "forgets structure" half of the Weatherall / Nguyen–Teh–Wells classification for every functor that records only gauge-invariant data, such as gauge orbits, field strengths and curvature conjugacy classes. Global (constant) gauge symmetries are always forgotten.