  We consider the exact channel synthesis problem. This problem concerns the
determination of the minimum amount of information required to create exact
correlation remotely when there is a certain rate of randomness shared by two
terminals. This problem generalizes an existing approximate version, in which
the generated joint distribution is required to be close to a target
distribution under the total variation (TV) distance measure (instead being
exactly equal to the target distribution). We provide single-letter inner and
outer bounds on the admissible region of the shared randomness rate and the
communication rate for the exact channel synthesis problem. These two bounds
coincide for doubly symmetric binary sources. We observe that for such sources,
the admissible rate region for exact channel synthesis is strictly included in
that for the TV-approximate version. We also extend the exact and
TV-approximate channel synthesis problems to sources with countably infinite
alphabets and continuous sources; the latter includes Gaussian sources. As
by-products, lemmas concerning soft-covering under R\'enyi divergence measures
are derived.
