classification of covering spaces

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

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