What I roughly know about time reversal operator T is,

$$

T[a] = a^\dagger, T[a^\dagger] = a\\

T[i] = -i, T[t] = -t.

$$

For the number operator $n=a^\dagger a$, I expect $T[n]=n$. Does that mean time reversal is NOT conjugating the field operator, which gives $T[a^\dagger a]=aa^\dagger$, but conjugating the whole thing? i.e. $T[a^\dagger a] = (a^\dagger a)^\dagger = a^\dagger a$?

But if that's the case, I don't understand $T[a|\psi(t)\rangle] = \langle\psi(-t)|a^\dagger$, since I expect $T[a|\psi(t)\rangle]=a^\dagger|\psi(-t)\rangle$.

Could somebody explain how to apply the time reversal to e.g. Heisenberg equation of motion?