Recasting straightening/unstraightening equivalence as $(\infty, 2)$-adjunction

Question

This is a vague set of questions that relies on (possibly non-existent) generalizations of low-dimensional results, mostly because I don't know many of the technical details underlying the constructions I will talk about.

Let $\mathrm{CAT}_\infty$ denote an $(\infty, 2)$-category of $(\infty,1)$-categories. There should be an $(\infty, 2)$-functor \begin{equation*} R : \mathrm{CAT}_\infty^{\mathrm{op}} \to \mathrm{CAT}_\infty \end{equation*} which sends $\mathcal{C}$ to the category of cocartesian fibrations over $\mathcal{C}$ and a functor $f : \mathcal{C} \to \mathcal{D}$ to the pullback functor along $f$. It is clear that $R$ preserves limits. If, furthermore, $R$ is a right adjoint, say with left adjoint $L$, then in particular we have the following chain of equivalences of $(\infty,1)$-categories: \begin{equation*} \mathrm{CoCart}(\mathcal{C}) = R(\mathcal{C}) \simeq \mathrm{Fun}(\ast, R(\mathcal{C})) \simeq \mathrm{Fun}^\mathrm{op}(L(\ast), \mathcal{C}) \simeq \mathrm{Fun}(\mathcal{C}, L(\ast)). \end{equation*} By setting $\mathcal{C} = \ast$ we see that $L(\ast) \simeq \mathrm{CoCart}(\ast) \simeq \mathrm{Cat}_\infty$, the $(\infty,1)$-category obtained by discarding all non-invertible $2$-morphisms in $\mathrm{CAT}_\infty$. Hence $\mathrm{CoCart}(\mathcal{C}) \simeq \mathrm{Fun}(\mathcal{C}, \mathrm{Cat}_\infty)$, which is a version of the straightening/unstraightening equivalence for $(\infty, 1)$-categories.

(Edit: I realized in the comments that if we assume straightening/unstraightening we can conclude, by a similar argument used to determine $L(\ast)$, that there is an adjunction $\mathrm{Fun}(-, \mathrm{Cat}_\infty) \dashv \mathrm{CoCart}$. So the two statements are equivalent. I am still interested in knowing which one is easier to prove.)

Now onto the questions:

1. How much of the theory of $(\infty,2)$-adjoints has been developed for this argument to make sense?
2. If the answer to the first equation is "enough", is there an adjoint functor theorem for $(\infty, 2)$-categories? The usual ones for $1$- or $(\infty, 1)$-categories require the source and target categories to be sufficiently nice (e.g., locally small, which is certainly not true in this case but can be replaced with something similar) and the requirements on $R$ are not terribly hard to check (preserves limits and certain kinds of colimits, see for example this paper). I would also like to know what $L$ looks like: is there a formula for left adjoints, when they exist?
3. Given 1. and 2., how much harder would this proof be compared to the usual proof of straightening/unstraightening (which takes a good deal of setup and model categorical machinery)? And most importantly, does the theory of $(\infty,2)$-categories depend unavoidably on straightening/unstraightening?
4. Obviously this does not only apply to the specific $R$ I picked: any right adjoint $\mathrm{CAT}_\infty^{\mathrm{op}} \to \mathrm{CAT}_\infty$ is representable with representing object $L(\ast)$, and something like it should be true for any self-enriched category. What are the conditions under which a representable functor is a right adjoint? This doesn't help answer the other questions, since we only know a posteriori that $\mathrm{CoCart}(-)$ is representable, but I am curious.

Answer 1

As suggested by Kevin Arlin, I am going to work out the $1$-categorical case below.

Recall that there are $2$-functors $\mathrm{Fun}(-, \mathrm{Cat}) : \mathrm{CAT} \to \mathrm{CAT}^\mathrm{op}$ and $\mathrm{OpFib} : \mathrm{CAT}^\mathrm{op} \to \mathrm{CAT}$, where $\mathrm{CAT}$ is the bicategory of large $1$-categories and $\mathrm{Cat}$ is the large $1$-category of small $1$-categories. I am going to restrict $\mathrm{OpFib}$ to spit out only opfibrations whose fibers are small categories. The goal is to construct a biadjunction $\mathrm{Fun}(-, \mathrm{Cat}) \dashv \mathrm{OpFib}$ by exhibiting a counit $\varepsilon : \mathrm{Fun}(\mathrm{OpFib}, \mathrm{Cat}) \Rightarrow 1_{\mathrm{CAT}^\mathrm{op}}$ and a unit $\eta : 1_{\mathrm{CAT}} \Rightarrow \mathrm{OpFib}(\mathrm{Fun}(-, \mathrm{Cat}))$ and proving that they satisfy the snake identities up to an invertible modification. The definition of biadjunction I am using is 2.1 in Gurski's paper Biequivalences in Tricategories (arxiv).

Counit: For each large category $C$ we will construct a functor $\varepsilon_C : C \to \mathrm{Fun}(\mathrm{OpFib}(C), \mathrm{Cat})$ (recall that we're in $\mathrm{CAT}^\mathrm{op}$, hence the change of orientation in the components of $\varepsilon$) that is natural in $C$, i.e. for all functors $F : C \to D$ we have $F_\ast \varepsilon_C \simeq \varepsilon_D F$. Since $$ \mathrm{Fun}(C, \mathrm{Fun}(\mathrm{OpFib}(C), \mathrm{Cat})) \simeq \mathrm{Fun}(\mathrm{OpFib}(C), \mathrm{Fun}(C, \mathrm{Cat})) $$ naturally in $C$ we can let $\varepsilon_C$ be the mate of the fiber functor which takes an opfibration $E \to C$ to the functor picking out its fibers (which are small categories by assumption) and a map $E \to E'$ of opfibrations over $C$ to the natural transformation whose components are the map restricted to each fiber. Now let $F : C \to D$ be a functor. Then \begin{align*} F_\ast \varepsilon_C (c) : \mathrm{OpFib}(D) \to \mathrm{Cat}, \quad & (E \to D) \mapsto (E \times_C D)_c, \\ \varepsilon_D F(c) : \mathrm{OpFib}(D) \to \mathrm{Cat}, \quad & (E \to D) \mapsto E_{F(c)}, \end{align*} and since $(E \times_C D)_c \simeq E_{F(c)}$ naturally in $c \in C$ this proves the required naturality condition.

Unit: For each large category $C$ we will construct a functor $\eta_C : C \to \mathrm{OpFib}(\mathrm{Fun}(C, \mathrm{Cat}))$ that is natural in $C$. For $c \in C$ let $T_c$ be the category whose objects are pairs $(F : C \to \mathrm{Cat}, a \in F(c))$ and morphisms $(F, a) \to (G, b)$ are pairs $(\alpha : F \Rightarrow G, h : \alpha_c(a) \to b)$. Note that $T_c$ comes with a functor $\pi$ to $\mathrm{Fun}(C, \mathrm{Cat})$. If $\alpha : F \to G$ is a natural transformation and $(F, a)$ is in the fiber over $F$, then $(\alpha, \mathrm{id}_{\alpha_c(a)}) : (F, a) \to (G, \alpha_c(a))$ is a cocartesian morphism in $T_c$ sitting over $\alpha$: for any $(H, z) \in T_c$, $$ \mathrm{Hom}_{T_c}((G, \alpha_c(a)), (H, z)) \cong \mathrm{Hom}_{T_c}((F, a), (H,z)) \times_{\mathrm{Nat}(F,H)} \mathrm{Nat}(G,H) $$ as can be verified by writing down the bijection explicitly. This proves that $\pi$ is an opfibration, so we can define $\eta_C(c)$ to be $\pi : T_c \to \mathrm{Fun}(C, \mathrm{Cat})$. On morphisms $f : c \to c'$ we do the obvious thing: there is a functor $T_c \to T_d$ sitting over $\mathrm{Fun}(C, \mathrm{Cat})$ which sends $(F, a)$ to $(F, F(f)(a))$ and it is easy to see that it preserves cocartesian morphisms, so it induces a map of opfibrations which we take to be $\eta_C(f)$. Now let $F : C \to D$ be a functor. Then \begin{align*} F_\ast \eta_C(c) & = T_c \times_{\mathrm{Fun}(C, \mathrm{Cat})} \mathrm{Fun}(D, \mathrm{Cat}), \\ \eta_D F(c) & = T_{F(c)}. \end{align*} There is a functor $T_{F(c)} \to T_c$ sending $(H : D \to \mathrm{Cat}, x \in H(F(c)))$ to $(HF : C \to \mathrm{Cat}, x \in (HF)(c))$, which together with the structure functor $T_{F(c)} \to \mathrm{Fun}(D, \mathrm{Cat})$ exhibits an isomorphism $T_{F(c)} \to T_c \times_{\mathrm{Fun}(C, \mathrm{Cat})} \mathrm{Fun}(D, \mathrm{Cat})$ of opfibrations over $\mathrm{Fun}(D, \mathrm{Cat})$ that is natural in $c$. This proves that $F_\ast \eta_C \simeq \eta_D F$ as functors.

Snake identities:

We need to prove that the composite transformation $$ \mathrm{OpFib} \Rightarrow \mathrm{OpFib}(\mathrm{Fun}(\mathrm{OpFib}, \mathrm{Cat})) \Rightarrow \mathrm{OpFib} $$ is equivalent to the identity modification. This can be done at the level of components, i.e. it will be enough to produce an equivalence of functors $\mathrm{OpFib}(C) \to \mathrm{OpFib}(C)$ for every $C$. The composition takes $(\rho : E \to C) \in \mathrm{OpFib}(C)$ to $\pi : T_\rho \to \mathrm{Fun}(\mathrm{OpFib}(C), \mathrm{Cat})$ (using the notation $c \mapsto T_c$ for the unit) and then pulls it back along the unit $C \to \mathrm{Fun}(\mathrm{OpFib}(C), \mathrm{Cat})$ to obtain an opfibration over $C$ whose fiber over $c$ is given by $$ (T_\rho)_{\mathrm{fib}_c} \simeq \{(\mathrm{fib}_c : \mathrm{OpFib}(C) \to \mathrm{Cat}, t \in \mathrm{fib}_c(\rho))\} \simeq \{t \in \rho^{-1}(c)\} = E_c; $$ this is evidently natural in $c$, so we're done with one identity. Another explicit calculation can be used to prove the other snake identity, asserting that the composition $$ \mathrm{Fun}(C, \mathrm{Cat}) \Rightarrow \mathrm{Fun}(\mathrm{OpFib}(\mathrm{Fun}(C, \mathrm{Cat})), \mathrm{Cat}) \Rightarrow \mathrm{Fun}(C, \mathrm{Cat}) $$ is equivalent to the identity functor of $\mathrm{Fun}(C, \mathrm{Cat})$ for any $C$. The composition takes a functor $H : C \to \mathrm{Cat}$ to the functor $\mathrm{fib}_H : \mathrm{OpFib}(\mathrm{Fun}(C, \mathrm{Cat})) \to \mathrm{Cat}$ which picks out the fiber over $H$, and then precomposes it with the unit to obtain the functor which is, on objects, $$ c \mapsto \mathrm{fib}_H(T_c) \simeq H(c), $$ where the equivalence is natural in $c$ by definition of $T_c$. And so we're done with the second identity, concluding the proof of the biadjunction.