  In the finite-dimensional case, we present a new approach to the theory of
cones with a mapping cone symmetry, first introduced by St{\o}rmer. Our method
is based on a definition of an inner product in the space of linear maps
between two algebras of operators and the fact that the Jamio{\l}kowski-Choi
isomorphism is an isometry. We consider a slightly modified class of cones,
although not substantially different from the original mapping cones by
St{\o}rmer. Using the new approach, several known results are proved faster and
often in more generality than before. For example, the dual of a mapping cone
turns out to be a mapping cone as well, without any additional assumptions. The
main result of the paper is a characterization of cones with a mapping cone
symmetry, saying that a given map is an element of such cone if and only if the
composition of the map with the conjugate of an arbitrary element in the dual
cone is completely positive. A similar result was known in the case where the
map goes from an algebra of operators into itself and the cone is a symmetric
mapping cone. Our result is proved without the additional assumptions of
symmetry and equality between the domain and the target space. We show how it
gives a number of older results as a corollary, including an exemplary
application.
