Functoriality algebras in lax monoidal functors

Construction 14.3.2. For an \(\infty \)-operad \(\Oo \), the construction \(C \mapsto \Alg _{\Oo }(C)\) defines a functor \[ \Alg _{\Oo }(-) := \Fun _{\Op _{\infty }}(\Oo ,-)\colon \Cat _{\infty }^{\otimes \text {-lax}} \to \Cat _{\infty }. \] Given a lax symmetric monoidal functor \(F\colon C \to D\), the induced functor \(\Alg _{\Oo }(C) \to \Alg _{\Oo }(D)\) is given by postcomposition with \(\Mm _C \to \Mm _D\). Its functoriality follows directly from functoriality of postcomposition in the defining fibers of functor categories.