  Given a monoidal adjunction, we show that the right adjoint induces a braided
lax monoidal functor between the corresponding Drinfeld centers provided that
certain natural transformations, called projection formula morphisms, are
invertible. We investigate these induced functors on Drinfeld centers in more
detail for the monoidal adjunction of restriction and (co-)induction along
morphisms of Hopf algebras. The resulting functors are applied to examples
related to affine algebraic groups, quantum groups at roots of unity, and
Radford--Majid biproducts of Hopf algebras. Moreover, we use the projection
formula morphisms to prove a characterization theorem for monoidal Kleisli
adjunctions and a crude monoidal monadicity theorem. The functor on Drinfeld
centers induced by the Eilenberg--Moore adjunction is given in terms of local
modules over commutative central monoids.
