  We categorify a class of skew-symmetrizable cluster algebras associated to
orbifolds using locally free representations of the gentle algebras arising
from orbifold triangulations. Along the way, we generalize
Derksen-Weyman-Zelevinsky's mutation on representations of quivers with
potentials to these gentle algebras and study the relation with $\tau$-tilting
theory. A Caldero-Chapton type formula concerning locally free
sub-representations is proven for cluster variables.
