Show naturality of $\infty$-natural transformation

Question

Working with the model category of complete Segal spaces $\text{CSS}$, which has as its underlying category the category of simplicial presheaves on $\Delta$, one has a suitable internal hom in $\text{CSS}$. Indeed, for complete Segal spaces $\mathcal{C}, \mathcal{D}$ one defines $\mathcal{D}^\mathcal{C} = \text{hom}(\mathcal{C} \times y_{\Delta^{\times 2}}, \mathcal{D})$ where $y_{\Delta^{\times 2}}$ is the Yoneda embedding $\Delta^{\times 2} \hookrightarrow \text{Psh}_\Delta(\Delta)$ into the category of simplicial presheaves on $\Delta$. Now the set $\mathcal{D}^\mathcal{C}_{0,0} = \text{hom}(\mathcal{C}, \mathcal{D})$ precisely corresponds to all $(\infty,1)$-functors and in the same spirit $\mathcal{D}^\mathcal{C}_{1,0} = \text{hom}(\mathcal{C}\times \Delta^1_{\bullet\star}, \mathcal{D})$ (where $\Delta^1_{\bullet\star}$ denotes the bisimplicial set which is constant in the second component, i.e., $(\Delta^1_{\bullet\star})_{k,l} = \Delta^1_k$) should correspond to the set of $(\infty,1)$-natural transformations.

Is there an elegant way to see that any $\zeta \in \text{hom}(\mathcal{C} \times \Delta^1_{\bullet\star}, \mathcal{D}) \cong \text{hom}(\mathcal{C}, \mathcal{D}^{\Delta^1_{\bullet\star}})$ really satisfies some naturality condition? More explicitly, we realize that $\zeta$ induces a map $$\zeta_{0,0} \colon \mathcal{C}_{0,0} \to \mathcal{D}^{\Delta^1_{\bullet\star}}_{0,0} \cong \mathcal{D}_{1,0}$$ i.e. $\zeta$ induces a family of 1-morphisms $(\zeta_c)_{c \in \mathcal{C}_{0,0}}$ in $\mathcal{D}$. I am expecting that this family of maps satisfies a naturality condition with respect to $\infty$-functors $F,G$ defined so that the diagram commutes. The mentioned naturality property should be so that the naturality square commutes, that is, $\zeta_{c'}\circ Ff \simeq Gf\circ\zeta_c$ for any $c, c' \in \mathcal{C}_{0,0}$ and any $(f \colon c \to c') \in \mathcal{C}_{1,0}$. Any ideas on this?

Answer 1

For every 1-morphism in $C$, i.e., a map $f\colon Δ^1→C$, the map $ζ$ induces a map $$ζ_f\colon Δ^1⨯Δ^1→D.$$

By the Yoneda lemma, the map $ζ_f$ is given by a pair of 2-simplices in $D$ that coincide along their 1st faces. Geometrically, it looks like a commutative square with a diagonal arrow in it.

This is precisely how naturality is implemented in complete Segal spaces: the naturality “square” is given by gluing two triangles along a common face. The data of triangles is additional data supplied to us by $ζ$, not just a property.

Additional higher coherence data for naturality is given by evaluating $ζ$ on higher simplices in $C$. These function in much the same way as 1-simplices.