A pseudomonoid in $(\mathbf{Cat}, 1, \times)$ is exactly a monoidal category. The pseudoaction of a monoidal category on a category is an actegory.

A pseudomonoid in $(\mathbf{Fib}, 1, \times)$ is a monoidal fibration. Pseudoactions of a monoidal fibration on a fibration are considered in (Vasilakopoulou ‘18).

A pseudomonad is a pseudomonoid in the Gray monoid of endomorphisms of an object in a Gray-category (see (Marmolejo ‘99, Definition 3.1)). A pseudoalgebra of a pseudomonad is a pseudoaction of the pseudomonad on the functor picking out the carrier of the pseudoalgebra.