  We study the asymptotic behaviour of the approximation, Gelfand and
Kolmogorov numbers of the compact embeddings of weighted function spaces of
Besov and Triebel-Lizorkin type in the case where the weights belong to a large
class. We obtain the exact estimates in almost all nonlimiting situations where
the quasi-Banach setting is included. At the end we present complete results on
related widths for polynomial weights with small perturbations, in particular
the sharp estimates in the case $\alpha=d(\frac 1{p_2}-\frac 1{p_1})>0$
therein.
