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The bijections of a countably infinite set form an uncountable set. Partition an enumeration into disjoint pairs and encode an infinite binary sequence by independently fixing or swapping each pair.

An uncountable set is a set for which no bijection with a subset of the natural numbers exists.

Two representations of a functor have unique mutually inverse morphisms carrying their specified universal elements to one another. Composing those morphisms preserves each universal element, so injectivity of the representing bijection makes the composites identities. Arbitrary isomorphisms of the underlying objects need not preserve the specified elements.