Monoidal functor

Definition

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ and $(\mathcal{D}, \otimes', I', \alpha', \lambda', \rho')$ be monoidal categories. A (lax) monoidal functor from $\mathcal{C}$ to $\mathcal{D}$ is a functor $F \colon \mathcal{C} \to \mathcal{D}$ together with:

1. a morphism $\varepsilon \colon I' \to F(I)$;
2. a natural transformation $\mu$ with components $\mu_{A,B} \colon F(A) \otimes' F(B) \to F(A \otimes B)$;

such that for all objects $A, B, C$ of $\mathcal{C}$:

1. associativity: $F(\alpha_{A,B,C}) \circ \mu_{A \otimes B, C} \circ (\mu_{A,B} \otimes' \id_{F(C)}) = \mu_{A, B \otimes C} \circ (\id_{F(A)} \otimes' \mu_{B,C}) \circ \alpha'_{F(A),F(B),F(C)}$;
2. left unitality: $F(\lambda_A) \circ \mu_{I,A} \circ (\varepsilon \otimes' \id_{F(A)}) = \lambda'_{F(A)}$;
3. right unitality: $F(\rho_A) \circ \mu_{A,I} \circ (\id_{F(A)} \otimes' \varepsilon) = \rho'_{F(A)}$.

The functor is strong monoidal if $\varepsilon$ and all $\mu_{A,B}$ are isomorphisms, and strict monoidal if they are identities.