  In the paper, we consider the harmonic maps between surfaces $\Sigma$ and $S$
in the homotopy class of a (branched) covering map $u_0$. We prove the
uniqueness of critical points of energy function and the injectivity of Hopf
differential if $u_0$ is a covering map. On the other hand, if $u_0$ is a
branched covering, we show that the uniqueness of critical points fails if
$u_0$ is a non-simple branched covering, and prove the injectivity of Hopf
differential $\Phi:\mc{T}(S)\to \op{QD}(\Sigma,g)$ when $g=[u_0^* h]$ for some
hyperbolic metric $h$ on $S$.
