Example 8.7.3.6. Following the convention of Remark 4.7.0.5, we say that a simplicial set is small if it is $\operatorname{\textnormal{\cjRL {t}}}$-small, for some fixed strongly inaccessible cardinal $\operatorname{\textnormal{\cjRL {t}}}$. Applying Proposition 8.7.3.5 in the special case $\kappa = \operatorname{\textnormal{\cjRL {t}}}$ and $\lambda = \operatorname{\textnormal{\cjRL {t}}}^{+}$, we deduce that there exists a cocompletion functor $T: \operatorname{\mathcal{QC}}_{< \operatorname{\textnormal{\cjRL {t}}}^{+} } \rightarrow \operatorname{\mathcal{QC}}_{< \operatorname{\textnormal{\cjRL {t}}}^{+} }$. The restriction of $T$ to small $\infty $-categories can be described more informally as follows:
For every small $\infty $-category $\operatorname{\mathcal{C}}$, the functor $T$ assigns the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ (see Theorem 8.4.0.3).
To every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ between small $\infty $-categories, $T(F): \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is left adjoint to the functor given by precomposition with $F^{\operatorname{op}}$ (see Remark 8.7.3.4 and Example 8.4.4.5).