Let $\mathcal E$ be an exact category, and let $K_0(\mathcal E)$ denote the Grothendieck group of $\mathcal E$, defined from isomorphism classes of objects modulo the relations coming from conflations. For every conflation
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\begin{align*}
0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0
\end{align*}
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in $\mathcal E$, one has
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\begin{align*}
[B]=[A]+[C]
\end{align*}
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in $K_0(\mathcal E)$.
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Moreover, let $G$ be an abelian group, and let $f:\operatorname{Ob}(\mathcal E)\to G$ be a function such that:
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1. if $X\cong Y$ in $\mathcal E$, then $f(X)=f(Y)$;
2. for every conflation
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\begin{align*}
0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0
\end{align*}
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in $\mathcal E$, one has
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\begin{align*}
f(B)=f(A)+f(C).
\end{align*}
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Then there exists a unique [group homomorphism](/page/Group%20Homomorphism)
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\begin{align*}
\overline f:K_0(\mathcal E)\to G
\end{align*}