$L^{\mathrm{\prime }}$-localization in an $\mathrm{\infty }$-topos

We prove that, given any reflective subfibration $L_{\bullet }$ on an $\mathrm{\infty }$-topos $\mathcal{E}$, there exists a reflective subfibration $L_{\bullet }^{\mathrm{\prime }}$ on $\mathcal{E}$ whose local maps are the $L$-separated maps, that is, the maps whose diagonals are $L$-local. This is the companion paper to “Localization theory in an $\mathrm{\infty }$-topos”.