Remark 9.2.1.2. The criterion of Proposition 9.2.1.1 is quite useful in practice, because the kinds of arguments used to verify conditions $(a)$ and $(b)$ often have a different flavor from one another:
To show that an $\infty $-category $\operatorname{\mathcal{C}}$ admits colimits of finite diagrams, we can reduce to the case of diagrams of very specific type: for example, $\operatorname{\mathcal{C}}$ is finitely cocomplete if and only if it admits pushouts and has an initial object (Corollary 7.6.2.30).
Many mathematical constructions preserve the formation of filtered colimits, but not general colimits. For this reason, it is often easy to check that some “forgetful” functor $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ creates filtered colimits (in the sense of Definition 7.1.4.18), so that the existence of filtered colimits in $\operatorname{\mathcal{C}}$ can be reduced to existence of filtered colimits in some (potentially simpler) $\infty $-category $\operatorname{\mathcal{D}}$.