Monoidal Functor

A functor F between two monoidal categories (C, ⊗, I) and (D, ⊗', I') equipped with a specified collection of structure maps: a natural transformation φ_{A,B}: F(A) ⊗' F(B) → F(A ⊗ B) and a unit map φ_0: I' → F(I), subject to coherence diagrams (associativity and unit constraints). Variants include strict, strong (invertible φ), lax, and oplax monoidal functors.