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

1. Suppose$$\begin{CD} 0@>>> A_1@>>> B_1@>>> C_1@>>> 0\\ & & @V\alpha VV @V\beta VV @V\gamma VV\\ 0@>>> A_2@>>>B_2@>>>C_2@>>>0 \end{CD$$ is a commutative diagram in $\mathcal{A}$ with exact rows. Then there is an exact sequence$$ 0 \to \ker\alpha \to \ker\beta \to \ker\gamma \stackrel{s}{\longrightarrow} \coker\alpha \to \coker\beta \to \coker\gamma\to 0,$$ usually called the kernel-cokernel sequence. The morphism $s$ is called the connecting morphism.
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$$ \cdots \to H_n(\mathbf{A})\to H_n(\mathbf{B})\to H_n(\mathbf{C}) \to H_{n-1}(\mathbf A)\to\cdots$$