  Centralized coded caching problem is studied for the two-user scenario,
considering heterogeneous cache capacities at the users and private channels
from the server to the users, in addition to a shared channel. Optimal caching
and delivery strategies that minimize the worst-case delivery latency are
presented for an arbitrary number of files. The converse proof follows from the
sufficiency of file-index-symmetric caching and delivery codes, while the
achievability is obtained through memory-sharing among a number of special
memory capacity pairs. The optimal scheme is shown to exploit the private link
capacities by transmitting part of the corresponding user`s request in an
uncoded fashion. When there are no private links, the results presented here
improve upon the two known results in the literature, namely, i) equal cache
capacities and arbitrary number of files; and ii) unequal cache capacities and
$N=2$ files. The results are then extended to the caching problem with
heterogeneous distortion requirements.
