  Let $C$ be an $m$-dimensional cone immersed in $\mathbb{R}^{n+m}$. In this
paper, we show that if $F:M^m \rightarrow \mathbb{R}^{n+m}$ is a properly
immersed mean curvature flow self-shrinker which is smoothly asymptotic to $C$,
then it is unique and converges to $C$ with unit multiplicity. Furthermore, if
$F_1$ and $F_2$ are self-expanders that both converge to $C$ smoothly
asymptotically and their separation decreases faster than
$\rho^{-m-1}e^{-\rho^2/4}$ in the Hausdorff metric, then the images of $F_1$
and $F_2$ coincide.
