  The recently discovered $T_{cc}^+$ is evaluated as a $DD^*$ molecular
structure in the $J^P=1^+$ sector. A coupled-channels calculation in charged
basis, considering the $D^0D^{*\,+}$, $D^+D^{*\,0}$ and $D^{*\,0}D^{*\,+}$
channels, is done in the framework of a constituent quark model that
successfully described other molecular candidates in the charmonium spectrum
such as the $X(3872)$. The $T_{cc}^+$ is found as a $D^0D^{*\,+}$ molecule
($87\%$) with a binding energy of $387$ keV/c$^2$ and a width of $81$ keV, in
agreement with the experimental measurements. The quark content of the state
forces the inclusion of exchange diagrams to treat indistinguishable quarks
between the $D$ mesons, which are found to be essential to bind the molecule.
The $D^0D^0\pi^+$ line shape, scattering lengths and effective ranges of the
molecule are also analyzed, which are found to be in agreement with the LHCb
analysis. We search for further partners of the $T_{cc}^+$ in other charm and
bottom sectors, finding different candidates. In particular, in the charm
sector we find a shallow $J^P=1^+$ $D^+D^{*\,0}$ molecule ($83\%$), dubbed
$T_{cc}^\prime$, just $1.8$ MeV above the $T_{cc}^+$ state. In the bottom
sector, we find an isoscalar and an isovector $J^P=1^+$ bottom partners, as
$BB^*$ molecules lying $21.9$ MeV/c$^2$ ($I=0$) and $10.5$ MeV/c$^2$ ($I=1$),
respectively, below the $B^0B^{*\,+}$ threshold.
