Given monoidal categories \((\mathcal{C}, \otimes, I_{\mathcal{C}})\) and \((\mathcal{D}, \bullet, I_{\mathcal{D}})\). A bifunctor \(F : \mathcal{C_1} \times \mathcal{C_2} \to \mathcal{D}\) is Semigroupal if it supports a natural transformation \(\phi_{AB,CD} : F\ A\ B \bullet F\ C\ D \to F\ (A \otimes C)\ (B \otimes D)\), which we call combine.

Laws

Associativity:

\[ \require{AMScd} \begin{CD} (F A B \bullet F C D) \bullet F X Y @>>{\alpha_{\mathcal{D}}}> F A B \bullet (F C D \bullet F X Y) \\ @VV{\phi_{AB,CD} \bullet 1}V @VV{1 \bullet \phi_{CD,FY}}V \\ F (A \otimes C) (B \otimes D) \bullet F X Y @. F A B \bullet (F (C \otimes X) (D \otimes Y) \\ @VV{\phi_{(A \otimes C)(B \otimes D),XY}}V @VV{\phi_{AB,(C \otimes X)(D \otimes Y)}}V \\ F ((A \otimes C) \otimes X) ((B \otimes D) \otimes Y) @>>{F \alpha_{\mathcal{C_1}}} \alpha_{\mathcal{C_2}}> F (A \otimes (C \otimes X)) (B \otimes (D \otimes Y)) \\ \end{CD} \]

combine . grmap combine . bwd assoc ≡ fmap (bwd assoc) . combine . glmap combine

Given monoidal categories \((\mathcal{C}, \otimes, I_{\mathcal{C}})\) and \((\mathcal{D}, \bullet, I_{\mathcal{D}})\). A bifunctor \(F : \mathcal{C_1} \times \mathcal{C_2} \to \mathcal{D}\) is Unital if it supports a morphism \(\phi : I_{\mathcal{D}} \to F\ I_{\mathcal{C_1}}\ I_{\mathcal{C_2}}\), which we call introduce.

Given monoidal categories \((\mathcal{C}, \otimes, I_{\mathcal{C}})\) and \((\mathcal{D}, \bullet, I_{\mathcal{D}})\). A bifunctor \(F : \mathcal{C_1} \times \mathcal{C_2} \to \mathcal{D}\) is Monoidal if it maps between \(\mathcal{C_1} \times \mathcal{C_2}\) and \(\mathcal{D}\) while preserving their monoidal structure. Eg., a homomorphism of monoidal categories.

See NCatlab for more details.

Laws

Right Unitality:

\[ \require{AMScd} \begin{CD} F A B \bullet I_{\mathcal{D}} @>{1 \bullet \phi}>> F A B \bullet F I_{\mathcal{C_{1}}} I_{\mathcal{C_{2}}}\\ @VV{\rho_{\mathcal{D}}}V @VV{\phi AB,I_{\mathcal{C_{1}}}I_{\mathcal{C_{2}}}}V \\ F A B @<<{F \rho_{\mathcal{C_{1}}} \rho_{\mathcal{C_{2}}}}< F (A \otimes I_{\mathcal{C_{1}}}) (B \otimes I_{\mathcal{C_{2}}}) \end{CD} \]

combine . grmap introduce ≡ bwd unitr . fwd unitr

Left Unitality:

\[ \begin{CD} I_{\mathcal{D}} \bullet F A B @>{\phi \bullet 1}>> F I_{\mathcal{C_{1}}} I_{\mathcal{C_{2}}} \bullet F A B\\ @VV{\lambda_{\mathcal{D}}}V @VV{I_{\mathcal{C_{1}}}I_{\mathcal{C_{2}}},\phi AB}V \\ F A B @<<{F \lambda_{\mathcal{C_{1}}} \lambda_{\mathcal{C_{2}}}}< F (I_{\mathcal{C_{1}}} \otimes A) (I_{\mathcal{C_{2}}} \otimes B) \end{CD} \]

combine . glmap introduce ≡ fmap (bwd unitl) . fwd unitl