During a course of mine time-reversal symmetry was introduced as an anti-linear operator. One property the lecturer pointed out is that \begin{equation} \langle\phi| A\psi\rangle \neq \langle A^\dagger \phi |\psi \rangle \end{equation} but rather we have \begin{equation} \langle\phi| A\psi\rangle = \langle \psi | A^\dagger \phi \rangle = \langle A^\dagger \phi | \psi \rangle^*. \end{equation} I understand that time-reversal involves a complex conjugation, but I don't understand what makes the above equalities true.
The question is: what property of time-reversal itself makes the above hold. Is it only complex conjugation or is there anything else needed for the above to be true?