A pseudo -algebra is the same as a lax algebra whose constraint 2-cells are invertible.
A pseudoalgebra is said to be normal or normalized if its unit constraint isomorphism is an identity.
While making a pseudoalgebra strict is quite difficult, usually making it normal is quite easy, and many pseudoalgebras arising naturally are normal. For instance, for the strict 2-monad whose strict algebras are strict monoidal categories and whose pseudoalgebras are unbiased non-strict monoidal categories, the unit constraint says that the “1-ary tensor product” is isomorphic to itself. Clearly in most cases it is most sensible to define the 1-ary tensor product to be , so that the pseudoalgebra is normal.
This situation is fairly general: if is a strict 2-monad for which the components of the unit are isocofibrations, then any pseudoalgebra structure can be modified to a normalized one on the same underlying object.
One way to state a coherence theorem is to say that every pseudoalgebra for a given 2-monad is equivalent to a strict one, perhaps in a structured way. See coherence theorems.