I'm reading this book "Introduction to Quantum Fields in Classical Backgrounds" by Mukhanov & Winitzki, and there in the chapter 8 "The Unruh Effect" they introduce 3 reference frames.
Laboratory Frame: "is the usual inertial reference frame with the coordinates $(t,x,y,z)$".
Proper Frame: "is the accelerated system of reference that moves together with the observer, we shall also call it the accelerated frame".
Comoving Frame: "defined at a time $t_0$ is the inertial frame in which the accelerated observer is instantaneously at rest at $t=t_0$. (Thus the term 'comoving frame' actually refers to a different frame for each $t_0$)".
Now, I don't understand why they say that the observer's proper acceleration at time $t=t_0$ is the 3-acceleration measured in the comoving frame at time $t_0$. Could you explain why? Also I don't understand completely the definition of comoving frame.