Warning 3.5.5.4. We have now given two a priori different definitions for the notion of $2$-groupoid:
According to Definition 2.2.8.24, a $2$-groupoid is a $2$-category $\operatorname{\mathcal{C}}$ such that every $1$-morphism of $\operatorname{\mathcal{C}}$ is an isomorphism and every $2$-morphism of $\operatorname{\mathcal{C}}$ is an isomorphism.
According to Definition 3.5.5.1, a $2$-groupoid is a simplicial set $X$ satisfying certain extension conditions.
We will show later that definitions are compatible: a simplicial set $X$ is a $2$-groupoid in the sense of Definition 3.5.5.1 if and only if it is isomorphic to the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is a $2$-groupoid in the sense of Definition 2.2.8.24 (in this case, the $2$-category $\operatorname{\mathcal{C}}$ is uniquely determined up to non-strict isomorphism; see Theorem 2.3.4.1). See Proposition 
.