Winges's description of lax monoidal functors

Theorem 16.2.2 (Winges (2026), Theorem 2.7). Let \(\Oo \) be an \(\infty \)-operad, and consider the induced functor \[ (s,t)\colon \Mon _{\Oo }(\Ar ^{\oplax }) \to \Mon _{\Oo }(\Cat _{\infty }) \times \Mon _{\Oo }(\Cat _{\infty }). \] Then:

(1)

The fiber over a pair \((C,D)\) of \(\Oo \)-monoidal \(\infty \)-categories is the \(\infty \)-category \(\Fun ^{\Oo \dlax }(C,D)\) of lax \(\Oo \)-monoidal functors \(C \to D\).

(2)

For fixed \(C\), the restriction of \((s,t)\) to \(\{C\} \times \Mon _{\Oo }(\Cat _{\infty })\) is a cocartesian fibration classifying \[ \Fun ^{\Oo \dlax }(C,-) \colon \Mon _{\Oo }(\Cat _{\infty }) \to \Cat _{\infty }. \]

(3)

For fixed \(D\), the restriction of \((s,t)\) to \(\Mon _{\Oo }(\Cat _{\infty }) \times \{D\}\) is a cartesian fibration classifying \[ \Fun ^{\Oo \dlax }(-,D) \colon \Mon _{\Oo }(\Cat _{\infty })\catop \to \Cat _{\infty }. \]