Proposition 23.2.10 (Coproducts and extensivity in cartesian unstraightenings). Let \(q\colon E \to B\) be a cartesian fibration, where \(B\) admits finite coproducts. If the straightening \[ \Str ^{\ct }(q)\colon B\catop \to \Cat _{\infty } \] preserves finite products, then \(E\) admits finite coproducts and \(q\) preserves them. More precisely, if \(X_a \in E_{b_a}\) and \(X \in E_{\coprod _a b_a}\) corresponds to \((X_a)_a\) under the equivalence \[ E_{\coprod _a b_a} \simeq \prod _a E_{b_a}, \] then the \(q\)-cartesian lifts \(X_a \to X\) of the coproduct inclusions exhibit \(X\) as the coproduct \(\coprod _a X_a\) in \(E\).
If, in addition, \(B\) is extensive, then \(E\) is extensive.
Proof. The first assertion is dual to Reference ? of [Cisinski et al. (2026)].
For the final assertion, write \(F := \Str ^{\ct }(q)\) and let \(Y := \coprod _a Y_a\), where \(Y_a \in E_{b_a}\). Since \(F\) preserves finite products, the coproduct of a finite family of \(q\)-cartesian morphisms is again \(q\)-cartesian: under the product equivalence between the relevant fibers, it corresponds to a tuple of isomorphisms. Hence the coproduct functor \[ \prod _a E_{/Y_a} \longrightarrow E_{/Y} \] is a cartesian functor over the extensivity equivalence \[ \prod _a B_{/b_a} \iso B_{/\coprod _a b_a}. \] By the dual of Reference ? of [Cisinski et al. (2026)], the induced functor on fibers over a tuple \((f_a\colon c_a \to b_a)_a\) is \[ \prod _a F(c_a)_{/f_a^*Y_a} \longrightarrow F\Bigl (\coprod _a c_a\Bigr )_{/(\coprod _a f_a)^*Y}. \] This is an equivalence because \(F\) preserves finite products. Hence the coproduct functor on slices is an equivalence, so \(E\) is extensive. □
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