Lax, strong, and strict monoidal functors

Let $(\mathcal{C},\otimes ,\mathbf{1},\alpha ,\lambda ,\rho )$ and $(\mathcal{D},⊠,{\mathbf{1}}^{\mathrm{\prime }},{\alpha }^{\mathrm{\prime }},{\lambda }^{\mathrm{\prime }},{\rho }^{\mathrm{\prime }})$ be monoidal categories (Monoidal category).

A lax monoidal functor is a functor $F:\mathcal{C}\to \mathcal{D}$ (Covariant functor, identity functor, composite functor, and contravariant functor) together with natural transformations (Natural transformation and its components)

$F_{2;X,Y}:F(X)⊠F(Y)⟶F(X\otimes Y),$

and

the morphism

$F_0:{\mathbf{1}}^{\mathrm{\prime }}⟶F(\mathbf{1}),$

such that, for all $X,Y,Z$,

$F({\alpha }_{X,Y,Z})\circ F_{2;X\otimes Y,Z}\circ (F_{2;X,Y}⊠1_{F(Z)})=F_{2;X,Y\otimes Z}\circ (1_{F(X)}⊠F_{2;Y,Z})\circ {\alpha }_{F(X),F(Y),F(Z)}^{\mathrm{\prime }},$

$F({\lambda }_X)\circ F_{2;\mathbf{1},X}\circ (F_0⊠1_{F(X)})={\lambda }_{F(X)}^{\mathrm{\prime }},$

$F({\rho }_X)\circ F_{2;X,\mathbf{1}}\circ (1_{F(X)}⊠F_0)={\rho }_{F(X)}^{\mathrm{\prime }}\mathrm{.}$

A lax monoidal functor is strong monoidal when the natural transformation $F_2$ is a natural isomorphism (Natural isomorphism)—equivalently, every component $F_{2;X,Y}$ is an isomorphism—and $F_0$ is an isomorphism.

A strong monoidal functor is strict monoidal when the functor preserves the unit and tensor on the nose and every structure map above is an identity.