Gamma$(2,2)$ prior plus Poisson data gives a Gamma$(6,5)$ posterior

Suppose $X_1,\dots,X_n$ are independent observations from a Poisson distribution with pmf $f(x\mid\lambda)=e^{-\lambda}\lambda^x/x!$, $x=0,1,2,\dots$, $\lambda>0$ unknown. Suppose the prior distribution of $\lambda$ is $g(\lambda)\propto\lambda e^{-2\lambda}$, $\lambda>0$. Let $n=3$ and the data $(X_1,X_2,X_3)=(1,1,2)$. Then, suppose we estimate $\lambda$ under squared error loss $L(\lambda,a)=(\lambda-a)^2$. Then, the Bayes estimator based on the given data is: