Corollary 5.1.2.3. Let $Q$ be a partially ordered set, let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(Q)$ be an inner fibration of $\infty $-categories, and let $g: Y \rightarrow Z$ be a morphism in $\operatorname{\mathcal{C}}$. Then $g$ is $q$-cartesian if and only if, for every object $X \in \operatorname{\mathcal{C}}$ satisfying $q(X) \leq q(Y)$, the map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { [g] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]
of Notation 4.6.9.15 is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.
Proof.
By virtue of Proposition 5.1.2.1, the morphism $g$ is $q$-cartesian if and only if, for each object $X \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes
5.4
\begin{equation} \begin{gathered}\label{diagram:trivial-cartesian-poset} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [r]^-{\theta _ X} \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(X), q(Y), q(Z) ) \times _{ \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(Y), q(Z)) } \{ q(g) \} \ar [r] & \operatorname{Hom}_{\operatorname{N}_{\bullet }(Q)}( q(X), q(Z) ) } \end{gathered} \end{equation}
is a homotopy pullback square. If $q(X) \nleq q(Y)$, then the Kan complexes on the left side of the diagram (5.4) are empty, so this condition is vacuous. If $q(X) \leq q(Y)$, then the Kan complexes on the lower half of the diagram are isomorphic to $\Delta ^{0}$, so that (5.4) is a homotopy pullback square if and only if $\theta _{X}$ is a homotopy equivalence (Corollary 3.4.1.5). We conclude by observing that, in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \times _{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \{ g\} \ar [rr] \ar [dr]_{\theta _{X}} & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [dl]^{ [g] \circ } \\ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z), & } \]
where the horizontal map is an isomorphism (Corollary 4.6.9.5).
$\square$