Lax symmetric monoidal functor

Definition 14.3.1. Let \(C\) and \(D\) be symmetric monoidal \(\infty \)-categories. A lax symmetric monoidal functor \(C \to D\) is a morphism of \(\infty \)-operads \(\Mm _C \to \Mm _D\). An oplax symmetric monoidal functor is a morphism of \(\infty \)-operads \(\Mm _{C\catop } \to \Mm _{D\catop }\). We denote by \[ \Cat _{\infty }^{\otimes \text {-lax}} \quad \subseteq \quad \Op _{\infty } \] the full subcategory spanned by the (multimorphism operads of) symmetric monoidal \(\infty \)-categories.

We write \[ \Fun ^{\otimes \text {-lax}}(C,D) := \Fun _{\Op _{\infty }}(\Mm _C,\Mm _D) \] for the \(\infty \)-category of lax symmetric monoidal functors from \(C\) to \(D\).