Example 4.6.6.12 (Pinched Morphism Spaces in the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category (Definition 2.2.1.1) and let $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ denote its Duskin nerve (Construction 2.3.1.1). Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, which we identify with vertices of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Using Example 4.6.1.12, we can identify $n$-simplices $\sigma $ of the simplicial set $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y)$ with commutative diagrams
in the category of $1$-morphisms $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. The image of the left-pinch inclusion morphism
consists of those simplices which correspond (under this identification) to commutative diagrams in which each of the leftward pointing $2$-morphisms $f_{i,j} \Rightarrow f_{i-1,j}$ is an identity map. In this case, the entire diagram is determined by the sequence of composable morphisms $f_{0,0} \Rightarrow f_{0,1} \Rightarrow f_{0,2} \Rightarrow \cdots \Rightarrow f_{0,n}$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. Similarly, the image of the right-pinch inclusion morphism
consists of those simplices which correspond to commutative diagrams in which the rightward pointing $2$-morphisms $f_{i,j} \Rightarrow f_{i,j+1}$ is an identity map, which ensures that the entire diagram is determined by the sequence of composable morphisms $f_{n,n} \Rightarrow f_{n-1,n} \Rightarrow f_{n-2,n} \Rightarrow \cdots \Rightarrow f_{0,n}$ in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. Allowing $[n] \in \operatorname{{\bf \Delta }}$ to vary, we obtain canonical isomorphisms of simplicial sets