Definition 17.2.1 (\(\Rr \)-cocartesian fibrations). Let \(S\) be an \(\infty \)-category and \(\Rr \subset S\) a wide subcategory. A functor \(p\colon E\to S\) is called \(\Rr \)-cocartesian if for every \(e\in E\) and every \(r\colon p(e)\to y'\) in \(\Rr \) there exists a \(p\)-cocartesian morphism \(\hat r_e\colon e\to e'\) lifting \(r\). A functor \(f\colon E\to E'\) over \(S\) between \(\Rr \)-cocartesian fibrations is \(\Rr \)-cocartesian if it preserves \(p\)-cocartesian morphisms over morphisms in \(\Rr \). We denote by \[ (\Cat _\infty )^{\Rr \mathrm {-cocart}}_{/S}\quad \subset \quad (\Cat _\infty )_{/S} \] the (non-full) subcategory spanned by the \(\Rr \)-cocartesian fibrations and \(\Rr \)-cocartesian functors.
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