Remark 9.1.4.6. Theorem 9.1.4.5 asserts that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between filtered $\infty $-categories is right cofinal if and only if, for every object $C \in \operatorname{\mathcal{C}}$ and every morphism $u: F(C) \rightarrow D$ in the $\infty $-category $\operatorname{\mathcal{D}}$, there exists a morphism $w: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$ and a $2$-simplex
of the $\infty $-category $\operatorname{\mathcal{D}}$. This is equivalent to the requirement that $F([w])$ factors through $[u]: F(C) \rightarrow D$ when regarded as a morphism in the homotopy category $\operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$. In particular, the condition that $F$ is right cofinal depends only on the induced functor of homotopy categories $\operatorname {h}\! \mathit{F}: \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname {h}\! \mathit{\operatorname{\mathcal{D}}}$.