Lemma 10.5.1 (Pointed realization). Every finite-product-preserving functor \(F\colon C\to D\) between \(\infty \)-categories with finite products induces a canonical morphism of pointed \(\infty \)-operads \[ F_*\colon (\Mm _{(C,\times )})_* \longrightarrow (\Mm _{(D,\times )})_*, \] whose color functor sends \((*\to X)\) to \((*\to F(X))\). If \(D\) admits finite colimits and its cartesian product preserves them separately in both variables, then its target is naturally equivalent to \(\Mm _{(D_*,\wedge )}\).
Proof. By Proposition 15.3.6, the functor \(F\) has a unique symmetric monoidal refinement for the cartesian monoidal structures. Applying the pointed-object construction \((-)_*\colon \Op _{\infty }^*\to \Op _{\infty }^{\pt }\) of Theorem 18.4.12 gives the asserted morphism. The identification of the target with the smash-product operad is Lemma 18.4.10. □
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