TITLE: The $\infty$-category of natural transformations as an end
QUESTION [7 upvotes]: Let $\mathcal{C}$ be an $\infty$-category viewed as a fibrant scaled simplicial set with all 2-simplices thin and let $\mathfrak{C}\!at_{\infty}$ be the $\infty$-bicategory of $\infty$-categories. A model for $\mathfrak{C}\!at_{\infty}$ is given by applying the marked version of the homotopy coherent nerve to $\operatorname{Set}_{\Delta}^{+}$.
We define the $\infty$-bicategory of functors $\operatorname{Fun}(\mathcal{C},\mathfrak{C}\!at_{\infty})$ as in the $(\infty,1)$-case. Thin 2-simplices are given by maps of simplicial sets $\mathcal{C}\times \Delta^2_{\sharp} \to \mathfrak{C}\!at_{\infty}$ where $\Delta^2_{\sharp} $ denotes the maximally scaled 2-simplex.
Given two functors $F,G: \mathcal{C} \to \mathfrak{C}\!at_{\infty}$ we can use Proposition 2.4 in here  to compute a model for $\operatorname{Nat}(F,G)$ in terms of a suitably lax slice construction.
I would like to show that $\operatorname{Nat}(F,G)\cong \lim_\limits{\operatorname{Tw}(C)}\operatorname{Fun}(F(-),G(-))$, which should obviously be true but I am having trouble constructing the universal cone for this functor.

REPLY [3 votes]: Let $\mathcal{C}$ be an $\infty$-category and $\mathbb{D}$ be an $\infty$-bicategory. In joint work with W. Stern (Enhaced twisted arrow categories )  we show that there is a natural equivalence of $\infty$-categories
$$\operatorname{Nat}(F,G) \simeq \lim_{\operatorname{Tw}(\mathcal{C})^{\operatorname{op}}} \operatorname{Map}_{\mathbb{D}}(F(-),G(-))$$
The "op" comes from the fact that we are using right fibration $\operatorname{Tw}(\mathcal{C}) \to \mathcal{C} \times \mathcal{C}^{\operatorname{op}}$.