  Given an algebraic theory which can be described by a (possibly symmetric)
operad $P$, we propose a definition of the \emph{weakening} (or
\emph{categorification}) of the theory, in which equations that hold strictly
for $P$-algebras hold only up to coherent isomorphism. This generalizes the
theories of monoidal categories and symmetric monoidal categories, and several
related notions defined in the literature. Using this definition, we generalize
the result that every monoidal category is monoidally equivalent to a strict
monoidal category, and show that the "strictification" functor has an
interesting universal property, being left adjoint to the forgetful functor
from the category of strict $P$-categories to the category of weak
$P$-categories. We further show that the categorification obtained is
independent of our choice of presentation for $P$, and extend some of our
results to many-sorted theories, using multicategories.
