  We show how our previous result based on the replica Bethe ansatz for the
Kardar Parisi Zhang (KPZ) equation with the "half-flat" initial condition leads
to the Airy$_2$ to Airy$_1$ (i.e. GUE to GOE) universal crossover one-point
height distribution in the limit of large time. Equivalently, we obtain the
distribution of the free energy of a long directed polymer (DP) in a random
potential with one fixed endpoint and the other one on a half-line. We then
generalize to a DP when each endpoint is free on its own half-line. It amounts,
in the limit of large time, to obtain the distribution of the maximum of the
transition process Airy$_{2\to 1}$ (minus a half-parabola) on a half line.
