The setting for the Universal Forcing theorem consists of admissible Law-of-Logic realizations. These realizations supply a carrier, a cost type, comparison operations, and laws ensuring identity, non-contradiction, excluded middle, composition, invariance, and nontriviality.
The theorem asserts that any two Law-of-Logic realizations have canonically equivalent forced arithmetic objects. These objects are initial Peano algebras, so the equivalence is the unique isomorphism between them.
The cited Lean anchors are universal_forcing, which states the meta-theorem, along with arithmeticOf defining the extracted arithmetic and arithmetic_invariant giving an equivalent formulation.