\begin{definition}[Definition:Exactness of Chain Complex at Object]
Let $\AA$ be an abelian category.
Let $\family {d_i : C_i \to C_{i - 1} }_{i \mathop \in \Z}$ be a chain complex in $\AA$.
Then $\family {d_i : C_i \to C_{i - 1} }_{i \mathop \in \Z}$ is exact at $C_j$ {{iff}} the canonical map $\Img {d_{i + 1} } \to \ker \paren {d_i}$ is an isomorphism.
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The canonical map is induced by Homomorphisms Theorem for Categories with Zero Object since by definition $d_{j} \circ d_{j+1} = 0$.
\end{definition}