Example 20.1.12. In the situation of Definition 20.1.10, assume that every cocartesian transport functor \(f_!\colon C_s \to C_t\) satisfies \(f_!(W_s) \subseteq W_t\). Then \(p\) is left derivable with respect to \(\{W_s\}_{s \in S}\). Indeed, the condition on \(f_!\) shows that \(\gamma _t \circ f_!\) uniquely factors through a functor \(\bL f_!\colon C_s[W_s^{-1}] \to C_t[W_t^{-1}]\), which by Example 20.1.8 is an absolute left derived functor. The second condition is then an immediate consequence of the universal property.
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