Corollary 17.2.10. The functor \(E_{\Rr }\) restricts to a functor \[ E_{\Rr }\colon (\Cat _\infty )^{\Ll \mathrm {-cocart}}_{/S}\longrightarrow \Cocart (S) \] which is left adjoint to the inclusion.
Proof. By Proposition 17.2.8, \(E_{\Rr }\) sends \(\Ll \)-cocartesian fibrations to cocartesian fibrations, showing that it restricts at the level of objects. The characterization of the cocartesian morphisms from Proposition 17.2.8 further shows that if \(f\colon E \to E'\) preserves \(\Ll \)-cocartesian morphisms, then \(E_{\Rr }(f)\) preserves cocartesian morphisms, showing that \(E_{\Rr }\) also restricts at the level of morphisms.
To show the resulting functor is left adjoint to the inclusion, we must show that the unit and counit of the adjunction from Proposition 17.2.7 lie in the appropriate subcategories. Given an \(\Ll \)-cocartesian fibration \(p\colon E \to S\), it is clear from the description of the cocartesian morphisms given in Proposition 17.2.8 that the unit \(i_{p}\colon E \to E_{\Rr }(p)\) preserves \(\Ll \)-cocartesian morphisms. Similarly, given a cocartesian fibration \(q\colon F \to S\), the counit \(\pi _q\colon E_{\Rr }(q) \to F\) preserves cocartesian morphisms: it sends a cocartesian morphism as displayed in Proposition 17.2.8 to the map \(r_!e \to r'_!e'\), which is \(q\)-cocartesian by the right cancellation property of \(q\)-cocartesian morphisms (see Lemma 23.1.14). □
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