  We give a new proof that three families of polynomials coincide: the double
Schubert polynomials of Lascoux and Sch\"utzenberger defined by divided
difference operators, the pipe dream polynomials of Bergeron and Billey, and
the equivariant cohomology classes of matrix Schubert varieties. All three
families are shown to satisfy a "co-transition formula" which we explain to be
some extent projectively dual to Lascoux' transition formula. We comment on the
K-theoretic extensions.
