Inertial frames are universal:
(Assumption) There is a one-to-one correspondence between the (x, y, z, t) tuples of any two inertial frames.
That is, at any one moment, an observer in an inertial frame encounters exactly one location and sees exactly one clock time of another inertial frame. And if two observers meet for just a moment, they will agree on the two tuples they perceive (i.e. their own one and that of the other's). Moreover, given an inertial frame K, anything that can be labelled by any other inertial frame is visible in K too, hence the term "universal".
Transitivity is assumed as well:
(Assumption) If two (x, y, z, t) tuples, of two inertial frames, both correspond to the same (x, y, z, t) tuple of a third inertial frame, they also correspond to each other.