  We characterize mutual information as the unique map on ordered pairs of
random variables satisfying a set of axioms similar to those of Faddeev's
characterization of the Shannon entropy. There is a new axiom in our
characterization however which has no analogue for Shannon entropy, based on
the notion of a Markov triangle, which may be thought of as a composition of
communication channels for which conditional entropy acts functorially. Our
proofs are coordinate-free in the sense that no logarithms appear in our
calculations.
