Corollary 4.4.2.2 (Duskin [MR1897816]). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then $\operatorname{\mathcal{C}}$ is a $2$-groupoid (in the sense of Definition 2.2.8.24) if and only if the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a Kan complex.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The $2$-category $\operatorname{\mathcal{C}}$ is a $2$-groupoid if and only if it is a $(2,1)$-category and the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a groupoid (Remark 2.2.8.25). The first condition is equivalent to the requirement that $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.3.2.1). If this condition is satisfied, then Corollary 2.3.4.6 supplies an isomorphism $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \simeq \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }$. The desired equivalence now follows from Proposition 4.4.2.1. $\square$