ScalingStacks

Proof. We first check that the construction of $\Upsilon$ induces a functor $\Cat^{\ex}_{\mathcal{T}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Cat_{\mathcal{S}}$. Let $f\colon{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ be a map of stable simplicial categories and write

$$f_{!}^{\mathrm{cf}}\colon\mathrm{Fun}_{\Delta}(\mathcal{C}^{\op},{\mathcal{S}})^{\mathrm{cf}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{Fun}_{\Delta}({\mathcal{D}}^{\op},{\mathcal{S}})^{\mathrm{cf}}$$

for the induced spectral functor. Suppose that $X\colon\mathcal{C}^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}$ is projectively cofibrant and fibrant and that $\mathrm{N}(X)\colon\mathrm{N}(\mathcal{C})^{\op}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{S}}_{\infty}$ is stably representable via the spectrum object $A=\{a_{i}\}$ in $\mathrm{N}\mathcal{C}$. Since the diagram

commutes (where the vertical maps are the stable Yoneda embeddings), we see that $f_{!}$ restricts to a spectral functor $\Upsilon({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Upsilon({\mathcal{D}})$.

To verify that $\Upsilon$ induces a simplicial functor, we must check that it preserves equivalences of stable simplicial categories. So suppose that $f\colon\mathcal{C}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{D}}$ is an equivalence of stable simplicial categories. Then it follows that $f_{!}^{\mathrm{cf}}$ is a DK-equivalence of spectral categories, as is its restriction to the stably representable objects. ∎