Kerodon

Remark 8.4.0.2. Stated more informally, condition $(2)$ of Definition 8.4.0.1 asserts that if $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor of $\infty $-categories where $\operatorname{\mathcal{D}}$ admits small colimits, then $f$ factors (up to isomorphism) as a composition $\operatorname{\mathcal{C}}\xrightarrow {h} \widehat{\operatorname{\mathcal{C}}} \xrightarrow {F} \operatorname{\mathcal{D}}$, where the functor $F$ preserves small colimits; moreover, this factorization is required to be essentially unique. In other words, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ should be “freely generated” by $\operatorname{\mathcal{C}}$ under small colimits.