Let $\mathcal{A}$ be an abelian category. The snake lemma consists of the following two claims:

1. 1.

   Suppose

   $$\begin{matrix}0&\cd@stack{\rightarrowfill@}{}{}&A_{1}&\cd@stack{\rightarrowfill@}{}{}&B_{1}&\cd@stack{\rightarrowfill@}{}{}&C_{1}&\cd@stack{\rightarrowfill@}{}{}&0\\ &&{\alpha}{\Big\downarrow}&&{\beta}{\Big\downarrow}&&{\gamma}{\Big\downarrow}&&\\ 0&\cd@stack{\rightarrowfill@}{}{}&A_{2}&\cd@stack{\rightarrowfill@}{}{}&B_{2}&\cd@stack{\rightarrowfill@}{}{}&C_{2}&\cd@stack{\rightarrowfill@}{}{}&0\end{matrix}$$

   is a commutative diagram in $\mathcal{A}$ with exact rows. Then there is an exact sequence

   $$0\to\mathrm{ker}\,\alpha\to\mathrm{ker}\,\beta\to\mathrm{ker}\,\gamma\stackrel{s}{\longrightarrow}\mathrm{coker}\,\alpha\to\mathrm{coker}\,\beta\to\mathrm{coker}\,\gamma\to 0,$$

   usually called the kernel-cokernel sequence. The morphism $s$ is called the connecting morphism.

2. 2.

   Applying the previous claim inductively, for any short exact sequence

   $$0\to\mathbf{A}\to\mathbf{B}\to\mathbf{C}\to 0$$

   of chain complexes in $\mathcal{A}$, there is a corresponding long exact sequence in homology

   $$\cdots\to H_{n}(\mathbf{A})\to H_{n}(\mathbf{B})\to H_{n}(\mathbf{C})\to H_{{n-1}}(\mathbf{A})\to\cdots$$