Fix an \(\infty\)-category \(\mathcal C\) with a factorization system \((\mathcal L,\mathcal R)\).
Assume that \(\mathcal C\) contains a terminal object. Then, there exists a left adjoint
to the fully faithful inclusion, which is given by the formula \(c \mapsto \mathrm{Fact}(c \rightarrow{\sf pt}_\mathcal C)\). Moreover, the right adjoint is the inclusion of the \(\mathcal L\)-local objects, and hence the left adjoint exhibits \(\mathcal C^\mathcal R\) as the localization \(\mathcal C[\mathcal L^{-1}]\).70Assume that \(\mathcal C\) is presentable and that \((\mathcal L,\mathcal R)\) is generated by a set \(S\) of morphisms in \(\mathcal C\). Then, the reflective localization ([00K9]) also identifies \(\mathcal C^\mathcal R\) with the (accessible) localization \(\mathcal C[S^{-1}]\).
Furthermore, if \(\mathcal C\) has a symmetric monoidal structure compatible with the factorization system and which has the terminal object as monoidal unit, then it induces a symmetric monoidal structure on \(\mathcal C^\mathcal R\), for which the left adjoint \(\mathcal C\rightarrow\mathcal C^R\) is symmetric monoidal.
Given a morphism \((\mathcal C_0,(\mathcal L_0,\mathcal R_0)) \xrightarrow{F} (\mathcal C_1,(\mathcal L_1,\mathcal R_1))\) in \(\mathrm{Cat}_\infty^{\text{f.s.},\mathcal L,\mathcal R}\) in which both \(\mathcal C_0\) and \(\mathcal C_1\) admit terminal objects and \(F({\sf pt}_{\mathcal C_0}) \simeq {\sf pt}_{\mathcal C_1}\), the reflective localizations of part ([00K8]) assemble into a morphism
of adjunctions.