  We study embeddings of Besov-type and Triebel-Lizorkin-type spaces, $id_\tau
: {B}_{p_1,q_1}^{s_1,\tau_1}(\Omega) \hookrightarrow
{B}_{p_2,q_2}^{s_2,\tau_2}(\Omega)$ and $id_\tau :
{F}_{p_1,q_1}^{s_1,\tau_1}(\Omega) \hookrightarrow
{F}_{p_2,q_2}^{s_2,\tau_2}(\Omega) $, where $\Omega \subset {\mathbb R}^d$ is a
bounded domain, and obtain necessary and sufficient conditions for the
compactness of $id_\tau$. Moreover, we characterise its entropy and
approximation numbers. Surprisingly, these results are completely obtained via
embeddings and the application of the corresponding results for classical Besov
and Triebel-Lizorkin spaces as well as for Besov-Morrey and
Triebel-Lizorkin-Morrey spaces.
