Corollary 3.3 (0NJZ) · ScalingStacks

0NJZ

Corollary 3.3. Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be small stable idempotent-complete $\infty$ -categories. There is an equivalence of small stable idempotent-complete $\infty$ -categories between $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ and $\infty$ -category of right-compact ${\mathcal{A}}^{\op}\widehat{\otimes}{\mathcal{B}}$ -modules.

0NK0

Proof. Since the Yoneda embedding is fully faithful, it suffices to look at the essential image of the composite. The image of $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ in $\mathrm{Fun}^{\ex}({\mathcal{A}}\widehat{\otimes}{\mathcal{B}}^{\op},{\mathcal{S}}_{\infty})$ under $\nu_{a}$ is identified with the image of $\mathrm{Fun}^{\ex}({\mathcal{A}},{\mathcal{B}})$ in $\mathrm{Fun}^{\ex}({\mathcal{A}},\mathrm{Fun}^{\ex}({\mathcal{B}}^{\op},{\mathcal{S}}_{\infty}))$ under the corresponding map ${\mathcal{S}}_{\infty}^{\omega}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{A}}$ . Since this lies inside the image of $B$ inside $\mathrm{Fun}^{\ex}({\mathcal{B}}^{\op},{\mathcal{S}}_{\infty})$ under the Yoneda embedding, the result follows. ∎