Confusion regarding the construction of in u-modes for a spacetime that has a horizon created in the finite past

I have been studying the following paper linked https://inspirehep.net/files/4d58177a3c19484ba097b48c9f4e9a7f where they are exploring Hawking Radiation in Dispersive Media. While constructing the theory and tools for the dispersionless model, around page 31, we construct the in u-modes through some manipulation of the already obtained u out-modes. Specifically, this section –

There, the analysis normally begins by assuming
that a horizon has not always existed, but forms at some finite time from a collapsing
spacetime. The in-modes, then, are well-defined, for they are constructed in a spacetime
where a horizon has not yet formed. It is found that the results of this analysis, in
which in-modes are defined as positive-norm with respect to the initial time coordinate,
are identical to the results obtained by ignoring the formation process and defining the
in-modes as positive-norm with respect to the Kruskal coordinate. Now, near x = 0, the
unit vector defined by the Kruskal coordinate is proportional, but opposite in direction,
to the unit vector defined by the x-coordinate. Therefore, purely positive-norm modes
with respect to the Kruskal coordinate must be analytic in the upper-half x-plane near
x = 0.

The above has been hard for me given I do not understand which time coordinate they speak of and then they introduce the Kruskal coordinate out of the blue. Till the point, the out and in character of the modes was defined by in which direction the wave packet moves on increasing the time value in its input but here I think they switched definitions which completely baffles me.

Also this is a long time after which I have posted a question so please be patient with me.