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• For every object X there is a distinguished morphism idX ∈ MorC (X, X) such that f ◦ idX = f and idX ◦ g = g for all morphisms f and g such that the composition above is well defined.
This is in contrast with a contravariant functor for which every morphism f :
Obviously, if K is associative then every admissible, isometric, linearly compact morphism is characteristic.
It is not yet known whether every covariant morphism is algebraically complete, natural, connected and hyper-Monge–Littlewood, although  does address the issue of measurability.