Remark 7.1.8.5. Let $B$ and $K$ be simplicial sets, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits both $B$-indexed colimits and $K$-indexed colimits. Stated more informally, Corollary 7.1.8.4 asserts that the colimit of a $F: B \times K \rightarrow \operatorname{\mathcal{C}}$ can be computed “one variable at a time”: that is, there are (canonical) isomorphisms
In particular, the formation of ($K$-indexed) colimits in $\operatorname{\mathcal{C}}$ is preserved by the formation of ($B$-indexed) colimits (see Corollary 7.3.8.8 for a closely related result). This can also be deduced from Corollary 7.1.4.28: the formation of $K$-indexed colimits is left adjoint to the diagonal map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$ (see Proposition 7.1.1.18), and therefore preserves $B$-indexed colimits (which are computed levelwise, by virtue of Proposition 7.1.8.2).
Beware that the formation of ($K$-indexed) colimits in $\operatorname{\mathcal{C}}$ is usually not preserved by ($B$-indexed) limits: we will return to this point in §7.7.7.