ScalingStacks

Proof. For $\mathcal{C}_{1},\mathcal{C}_{2}\in{st}$, we define their tensor product by

$$\mathcal{C}_{1}\otimes\mathcal{C}_{2}=(\operatorname{Ind}(\mathcal{C}_{1})\otimes\operatorname{Ind}(\mathcal{C}_{2}))^{c}$$

where the tensor product of the right hand side is calculated in the $\infty$-category $\mathcal{P}r^{\rm L}$ of presentable $\infty$-categories (with morphisms left adjoints), and the superscript ^c denotes the full $\infty$-subcategory of compact objects of a presentable $\infty$-category. Since $\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2}$ is idempotent complete and retracts of compact objects are compact, $(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2})^{c}$ is idempotent complete as well. Thus the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ is indeed an object of ${st}$.

For $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{D}\in{st}$, let $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ be the full $\infty$-subcategory of functors $\mathcal{C}_{1}\times\mathcal{C}_{2}\to\mathcal{D}$ that preserve finite colimits in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ separately. We claim that For $\mathcal{C}_{1},\mathcal{C}_{2}\in{st}$, the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ corepresents the functor $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},-)$ in the sense that for any $\mathcal{D}\in{st}$, there is a canonical equivalence

$$\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\simeq\Fun_{{st}}(\mathcal{C}_{1}\otimes\mathcal{C}_{2},\mathcal{D}).$$

As a consequence, the associativity and symmetry of the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ will immediately follow from the analogous properties of $\Fun^{\prime}$.

For $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{D}\in\mathcal{P}r^{\rm L}$, let $\Fun^{L\times L}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ be the full $\infty$-subcategory of functors $\mathcal{C}_{1}\times\mathcal{C}_{2}\to\mathcal{D}$ that preserve colimits in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ separately. To prove the claim, observe that the inclusion $\mathcal{D}\rightarrow\operatorname{Ind}\mathcal{D}$ induces a fully faithful functor

$$\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\rightarrow\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})\simeq\Fun^{L\times L}(\operatorname{Ind}\mathcal{C}_{1}\times\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})$$

Its essential image consists of functors that preserve compact objects. By definition of the monoidal structure on the $\infty$-category $\mathcal{P}r^{\rm L}$ of presentable $\infty$-categories, we have a further equivalence

$$\Fun^{L\times L}(\operatorname{Ind}\mathcal{C}_{1}\times\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})\simeq\Fun^{\rm L}(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D}).$$

Since the compact objects of $\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2}$ are generated by finite colimits of external products of compacts objects, we obtain an equivalence between $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ and the full $\infty$-subcategory of $\Fun^{\rm L}(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})$ consisting of functors that preserve compact objects. In other words, we have the asserted equivalence that characterizes the tensor product

$$\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\simeq\Fun_{{st}}((\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2})^{c},\mathcal{D}).$$

Finally, the assertion that the functor $\operatorname{Ind}:{st}\to\mathcal{P}r^{\rm L}$ is symmetric monoidal is immediate from the constructions and the natural equivalence $\operatorname{Ind}(\mathcal{C}^{c})\simeq\mathcal{C}$, for $\mathcal{C}\in\mathcal{P}r^{\rm L}$. ∎