Following nLab's page on monoidal adjunctions, it seems there is a bunch of options for an adjunction to become "monoidal" in some sense.
I think that the one called monoidal adjunction in the nLab page is usually presented as an adjunction in which the unit and counit are monoidal natural transformations. Similarly, the one called opmonoidal adjunction is usually presented as an adjunction in which the unit and counit are opmonoidal natural transformations.
I wonder if there is a similar characterisation for lax/colax monoidal adjunction in terms of its unit and counit?