Remark 5.4.5.11 (01XM)

Remark 5.4.5.11. To get a feeling for the content of Proposition 5.4.5.10, let us specialize to the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{\mathcal{D}})$ is the Duskin nerve of a $2$-category $\operatorname{\mathcal{D}}$. In this case, we can identify a $3$-simplex $\sigma : \Delta ^{3} \rightarrow \operatorname{\mathcal{C}}$ with a collection of objects $\{ X_ i \} _{0 \leq i \leq 3}$ of $\operatorname{\mathcal{D}}$, a collection of $1$-morphisms $\{ f_{ji}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq 3}$, and a collection of $2$-morphisms $\{ \mu _{kji}: f_{kj} \circ f_{ji} \Rightarrow f_{ki} \} $ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ f_{32} \circ (f_{21} \circ f_{10} ) \ar@ {=>}[rr]^-{\alpha }_-{\sim } \ar@ {=>}[d]_{ \operatorname{id}_{ f_{32}} \circ \mu _{210} } & & ( f_{32} \circ f_{21} ) \circ f_{10} \ar@ {=>}[d]^{ \mu _{321} \circ \operatorname{id}_{ f_{10} }} \\ f_{32} \circ f_{20} \ar@ {=>}[dr]_{ \mu _{320} } & & f_{31} \circ f_{10} \ar@ {=>}[dl]^{ \mu _{310} } \\ & f_{30} & } \]

is commutative, where $\alpha = \alpha _{f_{32}, f_{21}, f_{10} }$ is the associativity constraint for the composition of $1$-morphisms in $\operatorname{\mathcal{C}}$ (Proposition 2.3.1.9). The assumption that the outer faces of $\sigma $ are thin guarantees that the $2$-morphisms $\mu _{321}$ and $\mu _{210}$ are isomorphisms. In this case, Proposition 5.4.5.10 asserts that $\mu _{320}$ is an isomorphism if and only if $\mu _{310}$ is an isomorphism, which follows by inspection.