To reprise: we want to factor the environment into its individual objects and then describe TOAST as a composite of techniques for operating on the individual factors. We cannot properly define environments as Cartesian products of individual objects defined in isolation because we have no way of expressing actions involving multiple objects. We can, however, define a set of objects in the context of a minimal, schematic environment containing one copy of each object. Having done so, we now want to recapture the notion of an environment being some kind of product of objects of different types. We will do this by showing that an environment with two eggs can be thought of as two overlapping copies of an environment with one egg; the copies differ only in the choice of the egg.
We will treat environments as having state spaces formed as products of the state spaces of their objects. A state of the environment is a tuple of the states of its objects. A binding of the schematic environment to the real environment is a particular kind of projection from the complex environment to the schematic, one which is also a reduction. If all reasonable projections are valid bindings, then we will say the environment is uniformly reducible to the schematic environment.