Lemma 12.11.2. Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor between abelian categories. Then $F$ induces a homomorphism of $K$-groups $K_0(F) : K_0(\mathcal{A}) \to K_0(\mathcal{B})$ by simply setting $K_0(F)([A]) = [F(A)]$.

Proof. Proves itself. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).