We discuss a functorial construction that turns congruences of an algebra into new algebras and preserves commutators, TCT types, idempotent Malcev conditions, etc. This allows us to relativize structure results, like the characterization of supernilpotence in varieties that omit type 1, from algebras to congruences. It also reduces the subpower membership problem for finite algebras with cube term to subdirectly irreducible algebras with central monolith.