the functor "k" takes a ring "a" to the grothendieck group of the set of isomorphism classes of its finitely generated projective modules, regarded as a monoid under direct sum. any ring homomorphism "a" → "b" gives a map "k"("a") → "k"("b") by mapping (the class of) a projective "a"-module "m" to "m" ⊗ "b", making "k" a covariant functor.