Lemma 19.7.1. Interpreted one universe higher, the \(\infty \)-category \(\Cat _{\infty }^{\mathrm {coprod}}\) is presentable and semiadditive. Its zero object is the terminal category, and the cartesian product \(C\times D\) is also the coproduct of \(C\) and \(D\). Moreover, the tensor product \(\otimes _{\mathrm {coprod}}\) makes \(\Cat _{\infty }^{\mathrm {coprod}}\) presentably symmetric monoidal.
Proof. Presentability is the finite-coproduct case of [Lurie (2017), Lemma 4.8.4.2]. Finite products are computed in \(\Cat _{\infty }\) and therefore given by cartesian products of \(\infty \)-categories. The terminal category is also initial in \(\Cat _{\infty }^{\mathrm {coprod}}\), since a finite-coproduct-preserving functor from it to \(C\) must select an initial object of \(C\).
Let \(0_C\) and \(0_D\) denote initial objects. For every \(E\in \Cat _{\infty }^{\mathrm {coprod}}\), restriction along the functors \[ C\longrightarrow C\times D,\quad c\longmapsto (c,0_D), \qquad \text {and}\qquad D\longrightarrow C\times D,\quad d\longmapsto (0_C,d), \] induces an equivalence \[ \Fun ^{\amalg }(C\times D,E) \iso \Fun ^{\amalg }(C,E)\times \Fun ^{\amalg }(D,E). \] Indeed, the coproduct functor \(\amalg \colon E\times E\to E\), which is left adjoint to the diagonal and therefore preserves finite coproducts, supplies an inverse which sends a pair \((F,G)\) to the composite \[ C\times D\xrightarrow {F\times G}E\times E\xrightarrow {\amalg }E. \] The two composites are naturally isomorphic to the identity because every \((c,d)\) is the coproduct of \((c,0_D)\) and \((0_C,d)\) in \(C\times D\). Thus \(C\times D\) is also a coproduct in \(\Cat _{\infty }^{\mathrm {coprod}}\), proving semiadditivity.
Finally, the defining universal property of \(\otimes _{\mathrm {coprod}}\) exhibits tensoring with a fixed object as a left adjoint. It therefore preserves colimits, which proves the final assertion. □
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