Remark 15.1.10. For the morphism of adequate triples in the previous result, parts (2) and (3) of Theorem 15.1.7 follow from cancellation of \(p\)-cartesian morphisms and the definition of \(E_R\), respectively. For a \(p\)-cocartesian morphism \(\psi \colon U \to Y\) satisfying part (4), part (5) is precisely the invertibility of the BeckâChevalley transformations \[ g_!{l'}^* \longrightarrow l^*(p\psi )_!\colon E_{pU}\longrightarrow E_J \] for the pullback squares in part (4). Here \((p\psi )_!\) exists by taking \(J=pY\) and \(l=\id \) in that condition.
Proof of Theorem 15.1.3. Condition (1) of Definition 15.1.2 and Proposition 15.1.9 show that \((E,E_L^{p\dcart },E_R)\) is adequate and that \[ \Span (p)\colon \Span _{\ct ,R}(E)\longrightarrow \Span _{L,R}(C) \] is defined. Given a span \(X \xleftarrow {l} U \xrightarrow {r} Y\) and an object over \(X\), choose a \(p\)-cartesian lift of \(l\) followed by a \(p\)-cocartesian lift of \(r\). In the criterion of Theorem 15.1.7, parts (2) and (3) hold by the preceding remark, part (4) follows from condition (2) of Definition 15.1.2 because \(g\) is a base change of \(r\), and part (5) follows from condition (3) and Lemma 15.1.8. Hence the chosen span is cocartesian, with transport \(r_!l^*\).
Finally, by Lemma 13.1.13, a morphism in the fiber over \(X\) is represented by a span whose backwards leg is \(p\)-cartesian over \(\id _X\), hence an isomorphism, and whose forward leg is an arbitrary morphism of \(E_X\). Thus this fiber is \(E_X\). âĄ
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