Is the Stone-Čech compactification of the inverse limit of an inverse system $\left\{ X_{i},f_{ij},I\right\} $ of Tychonoff spaces equal to the limit of the inverse system $\left\{ \beta X_{i},\beta f_{ij},I\right\} $, where $\beta f_{ij}$ is the extension of $f_{ij}$ over $\beta X_{i}$ and $\beta X_{j}$?

Similarly, is the Hewitt realcompactification of the limit of an inverse system $\left\{ X_{i},f_{ij},I\right\} $ of Tychonoff spaces equal to the limit of the inverse system $\left\{ \upsilon X_{i},\upsilon f_{ij},I\right\} $, where $\upsilon X_{i}$ is the realcompactification of $X_{i}$?