ScalingStacks

Proof. We need to verify the hypothesis of 2.1.5. The underlying $\infty$-category functor $L\colon\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\to\mathbf{Cat}_{\infty}$ is a composition of two functors $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{un}}}\to\mathbf{Op}_{\infty}\to\mathbf{Cat}_{\infty}$. The first is a right adjoint by A.2.3.1.9 and the second is a right adjoint by A.2.1.4.10. Hence, the composition is a right adjoint as well and therefore preserves limits. Moreover, by 2.2.3, $L$ is also a left adjoint and its right adjoint is fully faithful. Hence, the functor $\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}\to\mathbf{Cat}_{\infty,*}$ also has a fully faithful right adjoint and we have $\left(\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}\right)^{\operatorname{\scriptsize{red}}}\simeq\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}$. Finally, by 2.1.5, the inclusion $\mathbf{Op}_{\infty}^{\operatorname{\scriptsize{red}}}\simeq\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{red}}}\hookrightarrow\mathbf{Op}_{\infty,*}^{\operatorname{\scriptsize{un}}}$ admits a right adjoint with the stated description. ∎