Variant 10.2.2.17. In the situation of Exercise 10.2.2.16, we can apply the nerve functor to (10.21) and obtain a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \{ a, d \} \ar [r] \ar [d]^{F_0} & \operatorname{N}_{\bullet }(Q) \ar [d]^{F} \\ \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \ar [r] & \Delta ^2, } \]

which we can regard as a pullback square in the $\infty $-category $\operatorname{\mathcal{QC}}$. Show that $F$ is a quotient morphism in $\operatorname{\mathcal{QC}}$, but that $F_0$ is not (beware that this is not a formal consequence of Exercise 10.2.2.16: the construction $P \mapsto \operatorname{N}_{\bullet }(P)$ does not preserve quotient morphisms in general).