If I take $\mathcal{A}$ = coherent sheaves on $X$* up to isomorphism, then there are two things I could do which come to mind.

The first is noticing that $(\mathcal{A},\oplus)$ is a monoid and subsequently applying Grothendieck's $K$-functor to it obtaining a group $K(\mathcal{A})$.

The second is to take the free abelian group generated by $\mathcal{A}$ quotiented out by relations

$B = A + C$ for any exact sequence $$0 \to A \to B \to C \to 0$$ obtaining a second group $L(\mathcal{A})$.

Are these two groups $K(\mathcal{A})$ and $L(\mathcal{A})$ isomorphic? If not, which one is the 'correct' one?

*I'm not sure what $X$ should stand for. I'd be happy to assume smooth and projective variety over $\mathbb{C}$, or maybe just locally noetherian scheme, or maybe just locally ringed space or etc.