Given a short exact sequence of chain complexes
$$0 \to \mathbf A \stackrel{\mathbf i}{\longrightarrow} \mathbf B \stackrel{\mathbf j}{\longrightarrow} \mathbf C \to 0$$
we get a long exact sequence of homology groups
$$\ldots \to H_n(\mathbf A) \stackrel{\mathbf i_*}{\longrightarrow} H_n(\mathbf B) \stackrel{\mathbf j_*}{\longrightarrow} H_n(\mathbf C) \stackrel{\partial}{\longrightarrow} H_{n-1}(\mathbf A) \stackrel{\mathbf i_*}{\longrightarrow} H_{n-1}(\mathbf B) \to \ldots$$
See Hatcher Theorem 2.16.
Now recall your question Chain homotopy definition . In the answer we have seen that there is a more general concept of chain complex allowing also non-zero groups in negative dimensions. Theorem 2.16 remains true for short exact sequences of such more general chain complexes; you do not even need to adapt the proof.
An example is the augmented singluar chain complex which has an additional $C_{-1}(X) = \mathbb Z$ (and $C_n(X) = 0$ for $n < -1$). Using this chain complex we get the long exact sequence for reduced homology groups.
The usual Mayer-Vietoris sequence is obtained by applying this to the short exact sequence
$$ 0 \to C_*(A \cap B) \stackrel{\varphi}{\longrightarrow} C_*(A) \oplus C_*(B) \stackrel{\psi}{\longrightarrow} C_*^{\mathcal U}(X) \to 0$$
with $\mathcal U = \{A,B\}$ and using the fact that $\iota : C_*^{\mathcal U}(X) \hookrightarrow C_*(X)$ is a chain homotopy equivalence (Proposition 2.21).
To get the Mayer-Vietoris sequence for reduced homology groups we shall show that Proposition 2.21 is also true for the augmented chain complexes (which we denote by $C_\#$); this means that the "augmented map" $\iota : C_\#^{\mathcal U}(X) \hookrightarrow C_\#(X)$ is a chain homotopy equivalence. What does this map look like in dimension $-1$?
Note that $C_0^{\mathcal U}(X) = C_0(X)$ because each singular $0$-simplex automatically maps into some $U_i \in \mathcal U$. Thus $\iota = id$ in dimension $0$ and we can augment $\iota$ with $id: \mathbb Z \to \mathbb Z$ in dimension $-1$.
On p. 123 Hatcher defines a chain map $\rho : C_*(X) \to C_*^{\mathcal U}(X)$ and a chain homotopy $D$ from $\mathbb I$ to $\iota \rho$ and moreover shows that $\rho \iota = \mathbb I$.
We conclude that $\rho = id$ in dimension $0$ because $\iota = id$ in dimension $0$. Therefore we can augment $\rho$ with $id: \mathbb Z \to \mathbb Z$ in dimension $-1$. These augented maps trivially satisfy $\rho \iota = \mathbb I$ on $C_\#^{\mathcal U}(X)$.
We next show that Hatcher's chain homotopy $D$ augments to a chain homotopy from $\mathbb I$ to $\iota \rho$ on $C_\#(X)$; this will prove our modification of Proposition 2.21. Recall that $D_{-1} : 0 \to C_0(X)$ for Hatcher's original $D$; thus $$\partial D_0 = \partial D_0 + D_{-1}\partial = id_0 - \iota_0 \rho_0 .$$
But $\iota_0 \rho_0 = id_0$, thus $\partial D_0 = 0$. Therefore we can augment $D$ as follows to $C_\#(X)$:
- Take $\bar D_{-1} : \mathbb Z \to C_0(X)$ and $\bar D_{-2} : 0 \to \mathbb Z$ the zero maps.
Then $\partial D_0 + \bar D_{-1}\varepsilon = 0 = id_0 - \iota_0 \rho_0$ and $\varepsilon \bar D_{-1} + \bar D_{-2}0 = 0 = id_{\mathbb Z} - \iota_{-1} \rho_{-1}$.
