classification of covering spaces

Let $X$ be a connected, locally path connected and semilocally simply connected space. Assume furthermore that $X$ has a basepoint $*$ .

A covering $p\colon\thinspace E\to X$ is called based if $E$ is endowed with a basepoint $e$ and $p(e)=*$ . Two based coverings $p_{i}\colon\thinspace E_{i}\to X$ , $i=1,2$ are called equivalent if there is a basepoint preserving equivalence $T\colon\thinspace E_{1}\to E_{2}$ that covers the identity, i.e. $T$ is a homeomorphism and the following diagram commutes