Prove that a simply connected covering space of X is also a covering space for any other covering space of X.
Actually I don't have an idea how to start with. But if X has a universal cover, then the covering space also have universal cover, then we need to prove that they are isomorphic?
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A simply connected covering space is called a universal covering space and the reason for calling it so is the fact you are asking a proof of.
Anyway, for a proof, use the lifting criterion. In this case it tells you that any map from a simply connected space to another space lifts to any covering space of the target space. Check that the lifting is a covering space. (You might need a locally path connected assumption on the base space )