Show naturality of $\infty$-natural transformation

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?

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.