UMVUE should be a function of sufficient statistic

Theorem) Rao-Blackwell theorem
Let $W$ be any unbiased estimator of $\tau(\theta)$, and $T$ be a sufficient statistic for $\theta$.
If we define $\phi(T)=E(W|T),$ then $E_\theta (\phi(T))=\tau(\theta)$ and $V_\theta (\phi(T)) \leq V_\theta(W)$
pf)

Theorem) Uniqueness of UMVUE
If $W$ is a UMVUE of $\tau(\theta),$ then $W$ is unique.
pf)

Theorem)
If $E(W)=\tau(\theta),$
$W$ is the UMVUE of $\tau(\theta)$ <=> $W$ is uncorrelated with all unbiased estimator of zero
pf)

How to construct UMVUE

Theorem) Lehmann-Scheffe theorem
If $T$ is CSS for $\theta$ and $E(\phi(T))=\tau(\theta),$ then
$\phi(T)$ is the MVUE for $\tau(\theta)$
pf)

• How to construct UMVUE when $T$ which is CSS is given
  ① Choose any unbiased estimator $W$ of $\tau(\theta)$
  ② Construct $E(W|T)$

ex1) $X_1,...,X_n \overset{iid}{\sim} Poisson(\lambda),$ UMVUE of $e^{-\lambda}=P(X_1=0)?$

ex2) $X_1,...,X_n \overset{iid}{\sim} Exp(\lambda),$ UMVUE of $e^{-\frac{1}{\theta}}=P(X_1>1)?$