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Remark Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $\{ \mu _{g,f} \} $ be a twisting cochain for $\operatorname{\mathcal{C}}$ (Notation, and let $\operatorname{\mathcal{C}}'$ be the twist of $\operatorname{\mathcal{C}}$ with respect to $\{ \mu _{g,f} \} $ (Construction Then the twisting cochain $\{ \mu _{g,f} \} $ defines a strictly unitary isomorphism of $2$-categories $\operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}'$, and therefore induces an isomorphism of simplicial sets $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}')$. In other words, the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ cannot detect the difference between $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}'$. This should be regarded as a feature, rather than a bug. Defining the composition law for $1$-morphisms in a $2$-category $\operatorname{\mathcal{C}}$ often requires certain arbitrary (but ultimately inessential) choices (see Example In such cases, one can often give a more direct description of the simplicial set $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ which avoids such choices. See Example and Corollary