Corollary 7.4.3.12. Let $F: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, let $\kappa $ be an uncountable regular cardinal for which $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small, and let $\lambda $ be a cardinal of exponential cofinality $\geq \kappa $. The following conditions are equivalent:
- $(1)$
The morphism $F$ is left cofinal.
- $(2)$
For every $\infty $-category $\operatorname{\mathcal{D}}$ and every limit diagram $\overline{G}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$, the composite map $( \overline{G} \circ F^{\triangleleft } ): \operatorname{\mathcal{C}}'^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is also a limit diagram.
- $(3)$
For every limit diagram $\overline{G}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the composite map $( \overline{G} \circ F^{\triangleleft } ): \operatorname{\mathcal{C}}'^{\triangleleft } \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ is also a limit diagram.
Proof.
The implication $(1) \Rightarrow (2)$ follows from Corollary 7.2.2.3 and the implication $(2) \Rightarrow (3)$ is immediate. We will complete the proof by showing that $(3)$ implies $(1)$. Assume that condition $(3)$ is satisfied; we wish to show that the map $F^{\operatorname{op}}: \operatorname{\mathcal{C}}'^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is right cofinal. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is a $\kappa $-small $\infty $-category. For every object $X \in \operatorname{\mathcal{C}}$, let $h_ X: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be the functor represented by $X$, set $G = h_{X}^{\operatorname{op}}$, and let $\overline{G}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow ( \operatorname{\mathcal{S}}^{< \kappa } )$ be an extension of $G$ to a limit diagram (so that $\overline{G}$ carries the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$ to a contractible Kan complex, by virtue of Example 7.4.3.7). By virtue of Proposition 7.4.3.11, it will suffice to show that the composite map
\[ \operatorname{\mathcal{C}}'^{\triangleleft } \xrightarrow { F^{\triangleleft } } \operatorname{\mathcal{C}}^{\triangleleft } \xrightarrow { \overline{G} } (\operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \]
is also a limit diagram. Our assumption that $\lambda \geq \mathrm{ecf}(\kappa )$ guarantees that the $\infty $-category $\operatorname{\mathcal{S}}^{< \kappa }$ is locally $\lambda $-small (Remark 5.5.4.14). Applying the criterion of Proposition 7.4.1.18, we are reduced to showing that for every representable functor $H: (\operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the composite map
\[ \operatorname{\mathcal{C}}'^{\triangleleft } \xrightarrow { F^{\triangleleft } } \operatorname{\mathcal{C}}^{\triangleleft } \xrightarrow { \overline{G} } (\operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \xrightarrow {H} \operatorname{\mathcal{S}}^{< \lambda } \]
is a limit diagram, which is a special case of assumption $(3)$.
$\square$