  We give an operadic definition of a genuine symmetric monoidal G-category,
and we prove that its classifying space is a genuine E_\infty G-space. We do
this by developing some very general categorical coherence theory. We combine
results of Corner and Gurski, Power, and Lack, to develop a strictification
theory for pseudoalgebras over operads and monads. It specializes to strictify
genuine symmetric monoidal G-categories to genuine permutative G-categories.
All of our work takes place in a general internal categorical framework that
has many quite different specializations.
  When G is a finite group, the theory here combines with previous work to
generalize equivariant infinite loop space theory from strict space level input
to considerably more general category level input. It takes genuine symmetric
monoidal G-categories as input to an equivariant infinite loop space machine
that gives genuine G-spectra as output.
