Example 8.3.4.22 (Spaces of Natural Transformation). Let $G,G': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories and let $\mathscr {H}$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. Combining Corollary 8.3.4.21 with Proposition 8.3.4.1, we obtain homotopy equivalences of Kan complexes
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})}( G, G' ) & \simeq & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {H} \circ (\operatorname{id}\times G), \mathscr {H} \circ (\operatorname{id}\times G') ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{S}})}( \underline{\Delta ^0}_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}, \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}}) }) \\ & \simeq & \varprojlim ( \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}}) }). \end{eqnarray*}
Stated more informally, the space of natural transformations from $G$ to $G'$ can be viewed as a limit of the diagram
\[ \operatorname{Tw}( \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}}\quad \quad (f: X \rightarrow Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(X), G'(Y)). \]