  Let \mu be any weight function defined on the unit disk $\Bbb D$ and let
$\phi$ be an analytic self-map of $\Bbb D$.
  In the present paper we show that the essential norm of composition operator
$C_\phi$ mapping from the $\alpha$-Bloch space, with $\alpha >0$, to
$\mu$-Bloch space $\mathcal{B}^\mu$ is comparable to $$ \limsup_{|a|\to
1^-}\|\sigma_a\circ\phi\|_{\mathcal{B}^{\mu}}, $$ where, for $a\in\Bbb D$,
$\sigma_a$ is a certain special function in $\alpha$-Bloch space.
