The inverse sends \(\Gg\) to the map \(\Gg_0 \to \abs{\Gg}\).
Proof
Consider the adjunction \(\Fun(\simp\catop,C) \rightleftarrows \Ar(C)\) from Lemma 2.31. By Lemma 2.32, the unit transformation is an isomorphism on groupoid objects. Consequently, the restriction of the realization functor to \(\Grpd(C)\) is fully faithful.Its essential image consists precisely of those morphisms \(f\colon U \to X\) for which the counit of the adjunction is an isomorphism. This counit takes the form of a commutative square and by definition this is an isomorphism if and only if \(f\) is an effective epimorphism in \(C\).
