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This question is a particular case of Tim Campion's question. Let $\mathrm{Adj}$ be the strict $2$-category corepresenting adjunctions, i.e., the free strict $2$-category generated by two objects $x, y$, two morphisms $l: x\to y$, $r: y\to x$, two $2$-cells $\eta: \operatorname{id}_x\to rl$, $\epsilon: lr\to \operatorname{id}_y$ and the triangle identity $(r\xrightarrow{\eta r} rlr\xrightarrow{r\epsilon} r)= \operatorname{id}_r$, $(l\xrightarrow{l\eta} lrl\xrightarrow{\epsilon l} l) = \operatorname{id}_l$.

Now we consider an $(\infty, 2)$-category $C$. For me, the adjunction here is by definition a functor $\mathrm{Adj}\to C$ (is it for most people?). I am wondering how much data I need to specify. It seems that people know from Riehl-Verity that I only need to specify the image of the above generator, subject to the triangle identity (which is now an equivalence of the $2$-cells), i.e., $\mathrm{Adj}$ is free as an $(\infty, 2)$-category with the above presentation. Examples include the following:

  1. In the $(\infty, 2)$-category of $(\infty, 1)$-categories, they say adjunctions are the same data as adjunctions in the homotopy $2$-category.
  2. In the proof of Theorem 4.6 of Rune Haugseng's paper, on pages 260 and 261, he seems to just provide the information on generators to define a functor from $\mathrm{Adj}$.

Scanning Riehl-Verity's paper I linked above, I could not find the exact same statement. Can someone clarify where I should look at?

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One has to unravel the language a little bit, but in Riehl-Verity you are looking for: Theorem 4.3.9, Theorem 4.3.11, Propositions 4.4.7, 4.4.11, 4.4.17, Theorem 4.4.18.

All of them are special cases of 4.4.7 so this is the one you should be looking at for the best statement, but "parental subcomputad" might be hard to parse, so I recommend having a look at all the other ones for very concrete, very parsable statements.

Note that the statement you gave is incorrect : if you provide the two triangle identities (i.e. homotopies filling yhe two triangles), you need to add one extra "higher triangle identity" to get something like Adj, otherwise there is some overdeterminacy. Or you can choose to remove one of the triangle identities, and only keep the property that the other one exists.

(This is similar to how you cannot specify the two adjoints without specifying one of the unit/counit, and if you specify both unit and counit, you have to specify one of the triangle identities etc.)

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    $\begingroup$ Worth adding that although the free $(\infty,2)$-category OP describes (call it $\newcommand{\Adj}{\mathrm{Adj}}\newcommand{\Adjwk}{\Adj_\mathrm{wk}}\Adjwk$) isn’t quite equivalent to $\Adj$ it’s true nonetheless that providing those generators suffices to present a map out of $\Adj$ — it’s just slightly overdetermined, as you say, so it’s not an equivalence but a retraction from $\Adjwk$ onto $\Adj$. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Sep 28, 2023 at 8:00
  • $\begingroup$ Thank you (the point you mentioned was a part of my confusion)! I think I'll accept this answer, but here's another part of the confusion: R-V paper states the theorem in terms of simplicial categories, and I was not sure if that takes care of all $(\infty, 2)$-categories. I guess it follows from generality that there is a model structure on simplicial categories where fibrant objects are quasi-category-enriched categories and cofibrations are simplicial subcomputad inclusions. Is it shown to be equivalent to $\mathrm{Cat}_{(\infty, 2)}$? $\endgroup$
    – nrkm
    Commented Sep 28, 2023 at 14:00
  • $\begingroup$ Ah, ok. I think that follows from the rectification of enriched $\infty$-categories (Theorem 5.8 of arxiv.org/pdf/1312.3881.pdf for V=Joyal model structure on sSet.). $\endgroup$
    – nrkm
    Commented Sep 28, 2023 at 14:27

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