Proposition 23.3.4 (Fiberwise initial objects in cartesian fibrations, [Lurie (2009), Proposition 2.4.4.9]). Let \(p\colon E \to C\) be a cartesian fibration whose fibers admit initial objects. Then the fiberwise initial objects assemble uniquely into a fully faithful left-adjoint section \[ s\colon C \to E \] of \(p\). Dually, the fiberwise terminal objects of a cocartesian fibration assemble uniquely into a fully faithful right-adjoint section.
Proof. For every \(x\in C\), choose an initial object \(e_x\in E_x\). We claim that \(e_x\), together with the identity morphism \(x\to p(e_x)=x\), is a left adjoint object to \(x\) under \(p\). Indeed, for any \(e\in E\) consider the map \[ \Hom _E(e_x,e)\longrightarrow \Hom _C(x,p(e)). \] For a morphism \(f\colon x\to p(e)\), choose a \(p\)-cartesian lift \(f^*e\to e\). The defining property of this lift identifies the fiber of the displayed map over \(f\) with \[ \Hom _{E_x}(e_x,f^*e), \] which is contractible because \(e_x\) is initial. It follows from Corollary 23.3.3 that \(\Hom _E(e_x,e)\to \Hom _C(x,p(e))\) is an equivalence. This proves the claim.
The pointwise criterion for adjunctions from Lemma 21.1.4 now assembles the objects \(e_x\) and the identity morphisms \(x\to p(e_x)\) into a left adjoint \(s\colon C\to E\) whose unit \(\id _C\to ps\) is a natural isomorphism. Hence \(s\) is a fully faithful section of \(p\). Any other section selecting initial objects in the fibers is a left adjoint to \(p\) by the same argument, and is therefore unique. The final statement follows by duality. □
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