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I would like to understand $\infty$ categorical adjunctions better. I am far from an expert, and so I would greatly appreciate published references (with no unproven foundational assumptions) wherever possible.

Let $S$ be a quasi-category and $p:\mathcal E \to S$ be a bi-cartesian fibration.

Morally, $p$ classifies a homotopy coherent functor $G: S \to \infty{\rm Cat}$ such that $G(\phi)$ admits a left adjoint for all edges $\phi \in S$. So, given an object $s \in S$, and an object $x \in \mathcal E_s$ we should be able to construct a functor ${\rm coun(p)}: S_{/s} \to \mathcal{E}_s$ by $$(\phi: s \to s') \mapsto F(\phi) \circ G(\phi)(x).$$

How does one construct this functor? More importantly how does one formally state that it is unique? I.e. how do we formalize that if we are given another bi-cartesian fibration $p'$ which classifies a a functor $G' :S \to \infty{\rm Cat}$ which is equivalent to $G$, then there is an equivalence ${\rm coun(p)} \simeq {\rm coun(p')}$

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