category with duals (list of them)
dualizable object (what they have)
Actually, since there are several types of monoidal functors (lax, colax, and strong) there are several types of “adjunctions between monoidal categories which respect the monoidal structure.” Namely, we could have:
An adjunction in the 2-category of monoidal categories and strong monoidal functors. In this case both the left and right adjoint are strong. We call this a strong monoidal adjunction.
An adjunction in the 2-category of monoidal categories and lax monoidal functors. In this case the right adjoint is lax, while the left adjoint is necessarily strong (by doctrinal adjunction; see here). This version, which is one of the most frequently occurring, is often called simply a monoidal adjunction.
The dual: an adjunction in the 2-category of monoidal categories and colax monoidal functors, in which case the left adjoint is colax and the right adjoint is strong. One might call this an opmonoidal adjunction.
A mixed situation, in which the left adjoint is colax, the right adjoint is lax, and the lax and colax structure maps are mates under the adjunction. This is a conjunction in the double category of monoidal categories and lax and colax monoidal functors, so we may call it a monoidal conjunction or a lax/colax monoidal adjunction. By doctrinal adjunction, given any adjunction between monoidal categories, if the right adjoint is lax monoidal, then the left adjoint automatically acquires a colax monoidal structure making the adjunction into a monoidal conjunction, and dually.