Uniqueness of the Comparison Functor

Just a hint....

First you should realize that

$$Φd=(Ud, h)$$

that is a T-algebra with underlying $Ud$, for any d in $D$. in order to obtain the structure h, observe that the 2 adjunctions have the same unit $\eta$, and deduce that $Φ\epsilon=\epsilon^T Φ$.

Then deduce that $\epsilon^TΦd=\epsilon^T(Ud, h)=h$ and so $Φ\epsilon=\epsilon^T Φ$ implies $U\epsilon_d=h$ so h is determined and $Φ$ is unique.