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Consider the (2,1) category $Cat$ of ($1$-)categories. There is a functor $$ Cat^{op}\times Cat\to Cat $$ sending $(C,D)$ to the functor category $Fun(C,D)$.

This gives rise to a fibration $F\to Cat^{op}\times Cat$. Objects of $F$ are triples $(C,D,f:C\to D)$, and a morphism $(C,D,f:C\to D)\longrightarrow (C',D',f':C'\to D')$ is a triple $(c:C'\to C,d:D\to D',f'\Rightarrow dfc)$.

I don't know a reference for this, but I'm convinced someone has done it in some way.

But what I am really looking for is an $(\infty,1)$-categorical version of this fibration, replacing the $(2,1)$-category $Cat$ with the $(\infty,1)$-category $Cat_\infty$ of $(\infty,1)$-categories. The twisted arrow category construction applied to $Cat_\infty$ would give something similar, but where $f'\Rightarrow dfc$ is a natural equivalence, while I'm looking for something where $f'\Rightarrow dfc$ can be any natural transformation (i.e. I'm interested in the functor $(C,D)\mapsto Fun(C,D)$, and not just in the maximal sub-$\infty$-groupoid version $(C,D)\mapsto Map_{Cat_\infty}(C,D)$).

Is there any reference where this is done/discussed? I want to use this construction; of course, I could do it myself, but I'd rather not re-invent the wheel.

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    $\begingroup$ As a partial answer, the specific case of $\mathrm{Fun}(-,X)$ for some fixed $X$ is treated Theorem 7.3 of arxiv.org/abs/1501.02161v3 $\endgroup$
    – daniel gratzer
    Commented Jun 23 at 14:40

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Section 7 of arxiv.org/pdf/2011.11042 discusses a general (op)lax twisted arrow category, which answers your question when applied to $Cat_{\infty}$. They cite arxiv.org/pdf/2009.11969 for introducing the op-lax version of the twisted arrow category.

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