  It is a classical result that any monoidal category is monoidally equivalent
to a strict one, called its strictification. In this paper, we prove that any
monoidal category is also monoidally equivalent to a non-strict monoidal
category, called its non-strictification. Furthermore, we show that these two
constructions give the free (non-)strict monoidal category generated by a
monoidal category. Moreover, we prove in detail that these two constructions
are part of a pair of free-forgetful 2-adjunctions.
