Left Adjoints for Generalized Multicategories

We construct generalized multicategories associated to an arbitrary operad in Cat that is $\mathrm{\Sigma }$-free. The construction generalizes the passage to symmetric multicategories from permutative categories, which is the case when the operad is the categorical version of the Barratt-Eccles operad. The main theorem is that there is an adjoint pair relating algebras over the operad to this sort of generalized multicategory. The construction is flexible enough to allow for equivariant generalizations.