Lemma 15.2.3. Let \(C\) be an \(\infty \)-category. Then the triple \((\Fin (C), \Fin (C)_{\ct }, \Fin (C))\) is an adequate triple. Furthermore, it is extensive in the sense of Definition 13.3.3.
Proof. The adequate triple is obtained by applying Proposition 15.1.9 to the cartesian fibration \(q\colon \Fin (C) \to \Fin \).
The cartesian straightening \(C^{(-)}\colon \Fin \catop \to \Cat _{\infty }\) of \(q\) preserves finite products by construction, while \(\Fin \) is extensive. The final assertion of Proposition 23.2.10 therefore shows that \(\Fin (C)\) is extensive.
The \(q\)-cartesian morphisms are closed under finite coproducts, and the morphisms \(\emptyset \to X\) and \(\nabla \colon X \sqcup X \to X\) are \(q\)-cartesian, by the componentwise description recalled above. The class of all morphisms has the same closure and containment properties automatically. Hence Lemma 13.3.4 shows that the adequate triple is extensive. □
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