Unfurling Beck Chevalley fibration

Theorem 15.1.3. Let \((C,C_L,C_R)\) be an adequate triple and let \(p\colon E \to C\) be a Beck–Chevalley fibration with respect to \((C_L,C_R)\). Set \(E_L^{p\dcart }:=p^{-1}(C_L)\cap E^{p\dcart }\) and \(E_R:=p^{-1}(C_R)\). Then \((E,E_L^{p\dcart },E_R)\) is an adequate triple, and the induced functor \[ \Span (p)\colon \Span _{\ct ,R}(E)\longrightarrow \Span _{L,R}(C) \] is a cocartesian fibration with fiber \(E_X\) over \(X\in C\). Its transport along a span \(X\xleftarrow {l}U\xrightarrow {r}Y\) is \(r_!l^*\).