Frequency of Reflected light in special relativity

[mancity]

Homework Statement

In a reference frame K a photon of frequency f falls normally on a mirror approaching it with relativistic velocity v. Find the momentum imparted to the mirror during the reflection of the photon (a) in the reference frame fixed to the mirror; (b) in the frame K.

Relevant Equations

f'/f = sqrt((1+(v/c))/(1-(v/c)))

In the. solution attached I'm not too sure why in frame K, we apply the doppler effect twice. Also, since the photon is moving away from the source, shouldn't the signs be switched? Thanks

 

[Ibix]

Your image is hard to read, but you seem to have the right answer in the mirror frame. So in that frame you have left- and right-going light of frequency $f'$. What frequency will those pulses have in $K$?