Kerodon

Remark 8.3.7.2. In the formulation of Theorem 8.3.7.1, we have implicitly assumed that the $\infty $-categories $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$ and $\operatorname{\mathcal{C}}$ are locally small (so that the functors $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{-} }$, $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{+} }$, and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}$ are well-defined). More generally, if we assume that $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}$ are locally $\lambda $-small (for some uncountable cardinal $\lambda $, which need not be small), then our proof of Theorem 8.3.7.1 will supply a pullback diagram (8.58) in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{\pm }, \operatorname{\mathcal{S}}_{< \lambda } )$.