A bijective linear (antilinear) operator $A$ on a Hilbert space $\mathcal{H}$ is called unitary (untiunitrary) if $\langle A\psi |A\phi \rangle =\langle \psi |\phi \rangle$ (resp. $\langle A\psi |A\phi \rangle =\langle \phi |\psi \rangle$) for all $\phi ,\psi \in \mathcal{H}$.
By the relation $\langle \psi |A^{\dagger}\phi \rangle =\langle A\psi |\phi \rangle$, for a unitary operator $A$ we can conclude that $\langle \psi |A^{\dagger}A\phi \rangle =\langle A\psi |A\phi \rangle =\langle \psi |\phi \rangle $ hence $A^{\dagger}A=I$. But how can I get that for an untiunitary operator $U$, we have $U^* U=I$?
If $\ U\ $ is antiunitary, then \begin{align} \langle\phi|\psi\rangle&=\langle U\psi|U\phi\rangle\\ &=\overline{\langle\psi|U^*U\phi\rangle}\\ &=\langle U^*U\phi|\psi\rangle \end{align} for all $\ \phi,\psi\ .$