Proposition 23.1.4 (Functoriality of partial cocartesian lifts). Let \(p\colon E\to C\) be a functor, and let \(W\subseteq \Ar (C)\) be a subcategory. Suppose that for every \((f,e)\in W\times _{s,C,p}E\) there exists a \(p\)-cocartesian lift of \(f\) starting in \(e\). Then these lifts assemble into a functor \[ \lift _W\colon W\times _{s,C,p}E\longrightarrow \Ar (E) \] More precisely, this functor factors through \[ \Ar (E)\times _{\Ar (C)\times _{s,C,p}E} \bigl (W\times _{s,C,p}E\bigr ), \] and the resulting functor is left adjoint to the projection from this pullback to \(W\times _{s,C,p}E\).
Proof. By Lemma 23.1.2, the required cocartesian lifts are precisely the left adjoint objects under the restricted functor \[ \Ar (E)\times _{\Ar (C)\times _{s,C,p}E} \bigl (W\times _{s,C,p}E\bigr ) \longrightarrow W\times _{s,C,p}E. \] The claim is therefore an immediate consequence of the pointwise criterion for adjunctions from Lemma 21.1.4. □
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