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

Laws

Associativity:

$$\require{AMScd} \begin{CD} (F A \bullet F B) \bullet F C @>>{\alpha_{\mathcal{D}}}> F A \bullet (F B \bullet F C) \\ @VV{\phi_{A,B} \bullet 1}V @VV{1 \bullet \phi_{B,C}}V \\ F (A \otimes B) \bullet F C @. F A \bullet (F (B \otimes C)) \\ @VV{\phi_{A \otimes B,C}}V @VV{\phi_{A,B \otimes C}}V \\ F ((A \otimes B) \otimes C) @>>{F \alpha_{\mathcal{C}}}> F (A \otimes (B \otimes C)) \\ \end{CD}$$

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 functor $F : \mathcal{C} \to \mathcal{D}$ is Unital if it supports a morphism $\phi : I_{\mathcal{D}} \to F\ I_{\mathcal{C}}$, which we call introduce.

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

See NCatlab for more details.

Laws

Right Unitality:

$$\require{AMScd} \begin{CD} F A \bullet I_{\mathcal{D}} @>{1 \bullet \phi}>> F A \bullet F I_{\mathcal{C}};\\ @VV{\rho_{\mathcal{D}}}V @VV{\phi A,I_{\mathcal{C}}}V \\ F A @<<{F \rho_{\mathcal{C}}}< F (A \otimes I_{\mathcal{C}}); \end{CD}$$

  combine . grmap introduce ≡ bwd unitr . fwd unitr

Left Unitality:

$$\begin{CD} I_{\mathcal{D}} \bullet F B @>{\phi \bullet 1}>> F I_{\mathcal{C}} \bullet F B;\\ @VV{\lambda_{\mathcal{D}}}V @VV{\phi I_{\mathcal{C}},B}V \\ F A @<<{F \lambda_{\mathcal{C}}}< F (A \otimes I_{\mathcal{C}} \otimes B); \end{CD}$$

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