Suppose that $T_{1}$ is sufficient and $T_{2}$ is minimal sufficient and let $U$ be an unbiased estimator of $\theta$ i.e., $\mathbb{E}(U)=\theta$. Define $U_{1}=\mathbb{E}(U|T_{1})$ and $U_{2}=\mathbb{E}(U|T_{2})$, since both $T_{1}$ and $T_{2}$ are sufficient, $U_{1}$ and $U_{2}$ are both independent of $\theta$ and are just functions of $T_{1}$ and $T_{2}$ respectively.
Show that $U_{2}=\mathbb{E}(U_{1}|T_{2})$
My Approach
Since $T_{2}$ is a minimal sufficient statistic, it can be expressed as a function of any other sufficient statistic, hence $T_{2}=f(T_{1})$, now since both $U_{1}$ are sufficient, we can write $U_{1}=g(T_{1})$ and $U_{2}=h(T_{2})=h(f(T_{1}))=k(T_{1})$, hence $\mathbb{E}(U_{2}|T_{2})=\mathbb{E}(U_{2}|T_{1})=U_{2}$, also $\mathbb{E}(U_{1}|T_{1})=U_{1}$
However, to find $\mathbb{E}(U_{1}|T_{2})=\mathbb{E}(g(T_{1})|f(T_{1}))$, I cant proceed beyond this.
Any help is much appreciated.