Let \(C\) and \(C'\) be accessible categories with pullbacks and weakly contractible colimits, and let \(G\colon C' \to C\) be a conservative functor which preserves pullbacks and weakly contractible colimits. If \(C\) is a locus, then also \(C'\) is a locus.
Proof
Consider a diagram \(X_{\bullet}\colon I \to C'\), with \(I\) weakly contractible, and set \(X := \colim_{i \in I} X_i\). We need to show that the top adjunction in the following diagram is an adjoint equivalence: By assumption, the bottom adjunction is an adjoint equivalence. Moreover, since \(G\) preserves pullbacks and weakly contractible colimits, the vertical maps (are defined and) commute with both of the adjoint functors. It then follows from conservativity of \(G\) that both the unit and counit of the top adjunction are isomorphisms, proving the claim.
