Corollary 2.3.2.7. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then every degenerate $2$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is thin.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Combine Theorem 2.3.2.5 with the observation that, for every $1$-morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the left and right unit constraints
\[ \lambda _{f}: \operatorname{id}_ Y \circ f \Rightarrow f \quad \quad \rho _{f}: f \circ \operatorname{id}_ X \Rightarrow f \]
are isomorphisms (in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Y)$). $\square$