Equivalence of almost canonical covers and simplicial finite-cover groupoids

Let $A$ be the underlying structure, and let $AC_A$ and $SFCG_A$ denote the categories introduced in the paper. Write

$$C: Iso(SFCG_A) \rightarrow Iso(AC_A)$$

for the functor between their isomorphism categories.

Equivalence conjecture. The categories $AC_A$ and $SFCG_A$ are equivalent. Moreover, this equivalence can be proved by extending the functor $C: Iso(SFCG_A) \rightarrow Iso(AC_A)$ to non-isomorphisms using a similar definition.

The claim concerns extending the established correspondence between almost canonical covers and simplicial finite-cover groupoids from isomorphisms to general morphisms. The supplied status evidence says that the categories are equivalent, so the conjectural claim is solved.