Let $X$, $Y$ be non-positively curved cube complexes and suppose there is a local isometry $X \to Y$. If $Y$ is special, then so is $X$.
This is Exercise 4.32 in M. Sageev's notes "CAT(0) Cube Complexes and Groups". It is somehow clear that this is true if one draws a picture, - but how can I prove this rigorously?
Thanks for any help!