Here we take an approach complementary to that in the book...one that does not use the spacetime diagrams in the same way but nevertheless obtains the same results.
Consider an observer with a light clock who observes the time for a tick of an identical light clock which moves relative to her with velocity V.
The following figure shows the set up:
The Pythagorean theorem suffices to express the needed relation between the time it takes to go from the bottom to the top mirror, the speed of the mirror with respect to O, and the vertical distance the light must travel:
Solving for tdu we find