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24 Chapter 4. The basic structure of morphisms in Loc
and
βα(h∗(b)) = εmeμh∗(b) = μεmeμh∗(b) = h∗hh∗(b) = h∗(b)
they are mutually inverse isomorphisms of complete lattices and hence, first,
h[L] is a locale.
Finally we will show that e (resp. m) has the frame homomorphism μβ
(resp. αε) for a left adjoint: we have
m((e(μβ))(x)) = meμεm(x) = hh∗(m(x)) ≥ m(x)
and hence e(μβ)(x) ≥ x (as m is a poset embedding), and μβ(e(a)) = μεme(a) = h∗h(a) ≤ a.
Similarly for m and αε.
