F multimorphism operad

Construction 17.3.11. We construct a functor \[ \Mm \colon \Mon _{\Ff }(\Cat _{\infty }) \to \Op _{\Ff }. \] Given an \(\Ff \)-monoidal \(\infty \)-category \(C\), its unstraightening \(p_{C}\colon C^{\otimes } \to \Span _{L,R}(F)\) is an \(\Ff \)-operad: conditions (1) and (2) are obvious, and (3) is proved just like in Lemma 14.2.4. Similarly, for an \(\Ff \)-monoidal functor \(C \to D\) (i.e. a morphism in \(\Mon _{\Ff }(\Cat _{\infty })\)) the map on unstraightenings \(C^{\otimes } \to D^{\otimes }\) is a cocartesian functor, hence in particular \(\Ll \)-cocartesian. This shows that unstraightening defines the desired functor \(\Mm \).