Remark 8.7.3.4. In the situation of Definition 8.7.3.3, suppose that $\eta : \operatorname{id}_{\operatorname{\mathcal{QC}}} \rightarrow T$ exhibits $T$ as a $\mathbb {K}$-cocompletion functor. Then, for every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between $\lambda $-small $\infty $-categories, the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^{F} \ar [d]^{\eta _{\operatorname{\mathcal{C}}} } & \operatorname{\mathcal{D}}\ar [d]^{ \eta _{\operatorname{\mathcal{D}}} } \\ T(\operatorname{\mathcal{C}}) \ar [r]^{T(F) } & T(\operatorname{\mathcal{D}}) } \]
commutes up to isomorphism. It follows that $T(F)$ can be identified with the $\mathbb {K}$-cocomplete extension of the functor $\eta _{\operatorname{\mathcal{D}}} \circ F$, in the sense of Remark 8.4.5.5.