Theorem 2.3.2.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\sigma $ be a $2$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$, corresponding to a diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z } \]
(see Example 2.3.1.15). Then $\sigma $ is thin if and only if $\gamma : g \circ f \Rightarrow h$ is an isomorphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$.