TITLE: Uniqueness of $\infty$-adjoints
QUESTION [5 upvotes]: Adjoints in a 2-category are essentially unique, in the following strong sense.  If $\mathbf{2}$ denotes the "walking arrow" category $(\cdot \to \cdot)$, then there is a 2-category $\mathrm{Adj}_1$ with two objects, and a map $\mathbf{2}\to \mathrm{Adj}_1$, such that for any 2-category $K$ the precomposition functor $\mathrm{Cat}(\mathrm{Adj}_1,K) \to \mathrm{Cat}(\mathbf{2},K)$ is fully faithful and its image consists of the 1-morphisms in $K$ with right adjoints.
Is an analogous fact true for adjoints in an $(\infty,2)$-category?  If so, what is the $(\infty,2)$-category $\mathrm{Adj}_{\infty}$?  A natural conjecture would be that $\mathrm{Adj}_{\infty} = \mathrm{Adj}_1$; I'm not quite sure whether to hope for that.
(I'm assuming that an "adjoint" in an $(\infty,2)$-category $K$ means a morphism whose image in the homotopy 2-category of $K$ has an adjoint in the usual sense.  So in a sense, this question is about "coherentification" of adjoints.)

REPLY [9 votes]: Yes, and $\mathrm{Adj}_\infty = \mathrm{Adj}_1$.  This is Theorem 4.4.18 of Riehl-Verity.  Thanks David, I should have remembered that.