Lemma 7.34.
The six-functor formalism descends further to a functor
\[D\colon \Span\left(\Shv_{\tau_D}^+(C), \Shv_{\tau_D^{\ad}}(C^{\ad})[\Sigma^{-1}_D]\right) \to \PrL.\]
Proof
By construction, \(D^!\) inverts \(\Sigma_D\). The span formalism is self-dual along admissible morphisms: the duality exchanges the pullback and lower-shriek operations. Hence every universal \(D^!\)-isomorphism is also a universal \(D^*\)-isomorphism. The four operations therefore descend through the two localizations, and the Beck–Chevalley and projection-formula transformations descend with them.