  In this paper, we continue our program of systematic categorification of the
Noncommutative Differential Geometry of Connes. We replace a ring with a small
$\mathbb C$-linear category, seen as a ring with several objects in the sense
of Mitchell. We introduced Fredholm modules over this category and construct a
Chern character taking values in the cyclic cohomology of $\mathcal C$. We show
that this categorified Chern character depends only on the homotopy class of
the Fredholm module and is well-behaved with respect to the periodicity
operator in cyclic cohomology.
