Grothendieck group of an essentially small abelian category

Let $\mathcal{C}$ be an essentially small abelian category (Abelian category). Its set of isomorphism classes is denoted $\mathrm{Iso}(\mathcal{C})$. The Grothendieck group of $\mathcal{C}$ is

$G_0(\mathcal{C}):=\mathbb{Z}[\mathrm{Iso}(\mathcal{C})]/⟨e_Y-e_X-e_Z:0\to X\to Y\to Z\to 0\text{ is short exact in }\mathcal{C}⟩\mathrm{.}$

where $\mathbb{Z}[\mathrm{Iso}(\mathcal{C})]$ is the free abelian group on that set (Free abelian group on a set) and $[X]$ denotes the image of the generator for the isomorphism class of $X$. Thus the defining relation is $[Y]=[X]+[Z]$ for every short exact sequence (Exact sequence and short exact sequence in an abelian category). This is the short-exact-sequence group, distinguished below from split Grothendieck groups of projectives. No free abelian group on the possibly class-sized collection of all objects is formed. The sequence $0\to 0\to 0\to 0$ in particular gives $[0]=0$.