We derive the quantum Teichm\"uller space, previously constructed by Kashaev
and by Fock and Chekhov, from tensor products of a single canonical
representation of the modular double of the quantum plane. We show that the
quantum dilogarithm function appears naturally in the decomposition of the
tensor square, the quantum mutation operator arises from the tensor cube, the
pentagon identity from the tensor fourth power of the canonical representation,
and an operator of order three from isomorphisms between canonical
representation and its left and right duals.