  We prove two results on the tube algebras of rigid C$^*$-tensor categories.
The first is that the tube algebra of the representation category of a compact
quantum group $G$ is a full corner of the Drinfeld double of $G$. As an
application we obtain some information on the structure of the tube algebras of
the Temperley-Lieb categories $TL(d)$ for $d>2$. The second result is that the
tube algebras of weakly Morita equivalent C$^*$-tensor categories are strongly
Morita equivalent. The corresponding linking algebra is described as the tube
algebra of the $2$-category defining the Morita context.
