nLab
monoidal structure map

For $F:C\to D$ a functor between categories that are equipped with the structure of monoidal categories $(C,\otimes )$, $(D,\otimes )$, a lax monoidal structure map is a natural transformation

$${\nabla }_{x,y}:F(x)\otimes F(y)\to F(x\otimes y)$$\nabla_{x,y}\colon F(x) \otimes F(y) \to F(x \otimes y)

that equips $F$ with the structure of a lax monoidal functor.

Similarly, an oplax monoidal structure map, or lax comonoidal structure map is a natural transformation

$${\Delta }_{x,y}:F(x\otimes y)\to F(x)\otimes F(y)$$\Delta_{x,y}\colon F(x \otimes y) \to F(x) \otimes F(y)

that equips $F$ with the structure of an oplax monoidal functor.

An (op)lax (co)monoidal structure map is sometimes called an (op)lax (co)monoidal transformation; however, this is not a laxification (a directed weakening) of any strong notion of monoidal natural transformation (which has nothing to laxify).