Let $\mathcal{C}$ be a small $\infty$-category. Using the straightening-unstraightening construction, we can define a hom-functor of $\mathcal{C}$ as a functor $\mathcal{C}(-,-):\mathcal{C}^{\mathrm{op}}\times \mathcal{C}\to \mathcal{S}$ classifying the twisted arrow category of $\mathcal{C}$. (The precise construction can be found in Construction 5.2.1.1 of Lurie's HA.) I wish to identify the hom-functor of $\operatorname{Fun}(\Delta^1,\mathcal{C})$ as a pullback of something more familiar. More precisely, I believe that there is a pullback square
$$\require{AMScd} \begin{CD} \operatorname{Fun}(\Delta^1,\mathcal{C})(-,-) @>{\operatorname{dom}}>> \mathcal{C}(\operatorname{dom}-,\operatorname{dom}-)\\ @V\operatorname{codom}VV @VVV \\ \mathcal{C}(\operatorname{codom}-,\operatorname{codom}-) @>>> \mathcal{C}(\operatorname{dom}-,\operatorname{codom}-) \end{CD}$$
in the $\infty$-category $\operatorname{Fun}(\operatorname{Fun}(\Delta^1,\mathcal{C}),\mathcal{S})$.
I know at least one ugly way to do this: Using the enriched Yoneda embedding, we can find a fully faithful functor $\mathcal{C}\to N(\mathbf{A}^\circ )$, where $\mathbf{A}$ is a combinatorial simplicial model category and $\mathbf{A}^\circ$ is its full simplicial subcategory spanned by the fibrant-cofibrant objects. We can then appeal to Proposition 4.2.4.4 of HTT to obtain an equivalence of $\infty$-cateogories $N((\mathbf{A}^{[1]})^\circ)\xrightarrow{\simeq} \operatorname{Fun}(\Delta^1,N(\mathbf{A}^\circ))$ (For definiteness, we shall give $\mathbf{A}^{[1]}$ the projective model structure.) The mapping space of $\mathbf{A}^{[1]}$ is given by the pullback square $$\require{AMScd} \begin{CD} \mathbf{A}^{[1]}(f,g) @>>> \mathbf{A}(\operatorname{dom}f,\operatorname{dom}g)\\ @VVV @VVV \\ \mathbf{A}(\operatorname{codom}f,\operatorname{codom}g) @>>> \mathbf{A}(\operatorname{dom}f,\operatorname{codom}g). \end{CD}$$
This square is homotopy cartesian if $f,g$ are projectively fibrant-cofibrant (For this means that $f,g$ are cofibrations between fibrant-cofibrant objects.), so this gives the desired identification of $\operatorname{Fun}(\Delta^1,\mathcal{C})(-,-)$. But it seems to me that I am using a sledgehammer to crack a tiny nut.
What I think is the right approach is to write the twisted arrow category of $\operatorname{Fun}(\Delta^1,\mathcal{C})$ as a homotopy pullback of relevant right fibrations over $\operatorname{Fun}((\Delta^1,\mathcal{C})\times \operatorname{Fun}(\Delta^1,\mathcal{C})^\mathrm{op}$ in the contravariant model structure. I was able to do this if we fix one variable, i.e., I was able to write down $\operatorname{Fun}(\Delta^1,\mathcal{C})_{/g}$ as a homotopy pullback of relevant right fibrations over $\operatorname{Fun}(\Delta^1,\mathcal{C})$ for a fixed morphism $g$ of $\mathcal{C}$. But I haven't been able to figure out the bivariant case. Does anyone know how to deal with the bivariant case with this approach?
I appreciate any comments and suggestions. I also welcome any other approaches to identify the hom-functor of $\operatorname{Fun}(\Delta^1,\mathcal{C})$. (I am interested in the bivariant hom-functors of the slice category $\mathcal{C}_{/x}$ for $x\in \mathcal{C}$, too. If anyone knows something about slices, please let me know!) Thanks in advance!
