= Solution
A longitudinal accelerating <electric field> changes longitudinal <momentum>, but exerts no direct transverse <force>. With a constant guiding <magnetic field> and adiabatic transport, the transverse gyro-energy is consequently retained from the cathode:
$$
\boxed{\Theta_\perp\simeq\Theta_c.}
$$
More generally the <magnetic moment> of the orbit gives $\Theta_\perp/B\simeq\text{constant}$, so the equality assumes the same relevant field strength, not merely that a field is present all along the line.
In the strong-magnetization regime, many <electron> gyro-orbits occur during an important long-range electron-ion interaction and the gyroradius is small. The <ion> then interacts predominantly with a <guiding centre>; rapid transverse cyclotron motion does not act like an unmagnetized bath at the full transverse <electron> <temperature>. This allows a <proton> equilibrium scale far below $T_c$ despite the hot transverse gyro-motion.<Theory of electron cooling by Yaroslav S. Derbenev>[magnetized-electron enhancement and equilibrium below transverse <electron> <temperature>]{ref}
That conclusion is conditional on magnetization and collision scales. A finite gyroradius, nonadiabatic fields or transverse beam errors can restore heating and degrade cooling; it is not generally true that transverse <temperature> can never affect the cooled beam.
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