Stabilization of lex operad is stable

Proof. Choose a universe in which \(D := \Env (\Oo )\) is small, let \(P := \PSh (D)\) be the corresponding presheaf category, and suppress the universe decoration from the notation. Let \[ \iota \colon \Oo \hookrightarrow \Mm _P \] be the multiplicative Yoneda embedding of Proposition 18.2.3. By Lemma 18.3.2, the operad \(\oSp (\Oo )\) is the full suboperad of \(\oSp (\Mm _P)\) spanned by those reduced excisive functors \(\An _*^{\fin } \to P\) whose values lie in the essential image of \(\Oo _{\lra {1}} \to P\).

By Corollary 18.2.4, the essential image of \(\Oo _{\lra {1}} \to P\) is closed under finite limits. Since \(\iota \) is fully faithful and preserves finite limits, it also reflects them. Thus reducedness and excisiveness can be checked either in \(\Oo _{\lra {1}}\) or after applying \(\iota \). Hence the essential image of the fully faithful functor \[ \Sp ^{\exc }(\Oo _{\lra {1}}) \to \Sp ^{\exc }(P) \] is closed under finite limits, since limits of reduced excisive functors are computed pointwise. The source and target of this functor are stable by Proposition 16.5.7, and a finite-limit-preserving functor between stable \(\infty \)-categories is exact. It follows that this essential image is closed under finite colimits as well. Since \(\oSp (\Mm _P)\) is stable by Lemma 18.4.2, the claim follows from Lemma 18.1.8. □