Exercise 7.2.5.11. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category (Definition 2.2.8.5). Show that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is a filtered $\infty $-category if and only if $\operatorname{\mathcal{C}}$ satisfies the following conditions:
The $2$-category $\operatorname{\mathcal{C}}$ is nonempty.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an object $Z \in \operatorname{\mathcal{C}}$ and a pair of $1$-morphisms $f: X \rightarrow Z$ and $g: Y \rightarrow Z$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and every pair of $1$-morphisms $f,g: X \rightarrow Y$, there exists a $1$-morphism $h: Y \rightarrow Z$ such that the $1$-morphisms $h \circ f$ and $h \circ g$ are isomorphic (when viewed as objects of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$).
For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ and every $2$-morphism $\gamma : f \Rightarrow f$, there exists a $1$-morphism $g: Y \rightarrow Z$ for which the horizontal composition $\operatorname{id}_{g} \circ \gamma $ is equal to the identity $2$-morphism $\operatorname{id}_{g \circ f}$.