Notation 9.4.1.19. Let $\mathbb {K}$ be a collection of simplicial sets, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be $\mathbb {K}$-cocomplete inner fibrations of simplicial sets. We let $\operatorname{Fun}^{\mathbb {K}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' )$ denote the full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}')$ spanned by those commutative diagrams
having the property that, for every vertex $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$.