Recall from Lemma 2.33 that sending a groupoid \(\Gg\) to the map \(\Gg_0 \to \bB \Gg\) defines an equivalence
\[\Grpd(T) \; \iso \; \EffEpi(T)\]
between the categories of groupoid objects and effective epimorphisms in \(T\). Moreover, it follows from descent that a morphism of groupoids \(f\colon \Hh \to \Gg\) is cartesian if and only if the induced square is a pullback square, giving an equivalence of categories \(\Grpd(T)^{\cart} \; \iso \; \EffEpi(T)^{\cart}\). Passing to slices over \(\Gg\), this results in an equivalence
Here the last equivalence is induced by the target map. Restricted to cartesian morphisms over the fixed effective epimorphism \(\Gg_0\to\bB\Gg\), this functor is an equivalence: its inverse sends a morphism \(B \to \bB\Gg\) to the pullback square
