The way to think about this is to first consider ordinary matter, and then apply what we know to be true for such bodies to light.
In cases where ordinary matter would normally absorb energy by speeding up, light will turn bluer. In cases where ordinary matter would normally loose energy and slow down, light will turn redder.
Since a ball tossed towards us from an oncoming object will move faster than a ball tossed towards us from a receding body, we know that photons reflected by a body approaching us will be bluer than photons reflected by a receding body.
Green photons reflected by body moving
from right to left
It should also be noted that it does not matter whether it is the observer or the observed object that moves.
If the observer travels towards an object, the photons will be registered as bluer than if the observer travels away from the object.
Green photons registered by detector moving from right to left
Red-shift and blue-shift of light is therefore entirely about relative motion. It does not matter what energy level the photon has while traversing the space between the observer and the observed. All that matters is relative motion. If the observer and the observed travel at identical speed, no net red-shift will be detected, even if both travel at a tremendous speed relative to some other object.
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