Lax Monoidal Functor

A lax monoidal functor between two Monoidal Categories $(\mathcal{C},\oplus,I)$ and $(\mathcal{D},\otimes,J)$ is a Functor $\mathcal{C}\to\mathcal{D}$ together with

• A morphism $J\to F(I)$
• A Natural Transformation $\mu_{X,Y}:F(X)\otimes F(Y)\to F(X\oplus Y)$

Subject to some coherence conditions.

References

• https://ncatlab.org/nlab/show/monoidal+functor