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 $p:E\to X$ is called based if $E$ is endowed with a basepoint $e$ and $p(e)=*$. Two based coverings $p_i:E_i\to X$, $i=1,2$ are called equivalent if there is a basepoint preserving equivalence $T:E_1\to E_2$ that covers the identity, i.e. $T$ is a homeomorphism and the following diagram commutes