Proof.
For any small \(\infty\)-category \(\mathcal C\), the presheaf category \(\mathcal P(\mathcal C)\) is generated by a small set of tiny objects, i.e. objects \(c\in \mathcal C\) for which \(\mathrm{Hom}_{\mathcal C}(c,-)\colon \mathcal C\rightarrow\mathcal S\) preserves all small colimits; such a small set of tiny objects is for example provided by the objects of \(\mathcal C\) itself. Moreover, any functor \(F\colon \mathcal C\rightarrow\mathcal D\) induces a left adjoint functor \(\mathcal P(F)\colon \mathcal P(\mathcal C) \rightarrow\mathcal P(\mathcal D)\) which preserves tiny objects. By an argument entirely analogous to the proof of corollary 3.2.6, it follows that \(\mathcal P(\mathcal C)\) is in particular projectively generated, and that \(\mathcal P(F)\) preserves compact-projectives. Hence, the symmetric monoidal functor \(\mathcal P\colon \mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^\mathrm{L}\) factors through the symmetric monoidal subcategory \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) from proposition 3.2.10. Moreover, the functor \(\mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\) is a left adjoint since after composing with the equivalence \(\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\simeq \mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) from proposition 3.2.8 it becomes equivalent to the left adjoint \(\mathrm{Cat}_{\infty}\rightarrow\mathrm{Cat}_{\infty}^{\sqcup, \mathrm{idem}}\) of the forgetful functor. Since moreover \(\mathcal S\rightarrow\mathrm{Cat}_{\infty}\) is a left adjoint, so is the composite \(\mathcal S\rightarrow\mathrm{Cat}_{\infty}\rightarrow\mathrm{Pr}^{\mathrm{L},\mathrm{cp}}\). ◻