  We consider the Gelfand and Kolmogorov numbers of compact embeddings between
weighted function spaces of Besov and Triebel-Lizorkin type with polynomial
weights in the non-limiting case. Our main purpose here is to complement our
previous results in \cite{ZF10} in the context of the quasi-Banach setting, $0
< p, q \le \infty$. In addition, sharp estimates for their approximation
numbers in several cases left open in Skrzypczak (JAT, 2005) are provided.
