$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary (Duskin [MR1897816]). Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then $\operatorname{\mathcal{C}}$ is a $2$-groupoid (in the sense of Definition if and only if the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a Kan complex.

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 The first condition is equivalent to the requirement that $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem If this condition is satisfied, then Corollary 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 $\square$