Prove the isomorphism I am trying to prove the following:
$$\left[S^1 \vee S^1 \vee ... \vee S^1; S^1\right] \cong \mathbb{Z}^n$$ 
where $\vee$ is a wedge sum and $[X,Y]$ is homotopy class between $X$ and $Y$.
I would appreciate a clear proof of this statement.
 A: Since $\bigvee^n_{i = 1} S^1 = * \cup_{\bigsqcup^n_{i = 1} \varphi} \bigsqcup^n_{i = 1} D^1 = C(\bigsqcup^n_{i = 1} \varphi)$ where $\varphi$ attaches $D^1$ along $\partial D^1 = S^0$ to $*$ because of this we can write down following cofibration sequence:
$$\bigsqcup^n_{i = 1}S^0 \xrightarrow{\bigsqcup^n_{i = 1} \varphi}* \to \bigvee^n_{i = 1} S^1$$
and we may use contravariant Puppe sequence:
$$\ldots\to\left[\Sigma*, S^1\right]\to\left[\Sigma\bigsqcup^n_{i = 1}S^0, S^1\right]\to\left[\bigvee^n_{i = 1} S^1, S^1\right]\to\left[*, S^1\right]\to\left[\bigsqcup^n_{i = 1}S^0, S^1\right]
$$
ofcourse
$\left[\Sigma*, S^1\right], \left[*, S^1\right] = 0$ hence
$\left[\Sigma\bigsqcup^n_{i = 1}S^0, S^1\right]\simeq\left[\bigvee^n_{i = 1} S^1, S^1\right]$ but $\left[\Sigma\bigsqcup^n_{i = 1}S^0, S^1\right] = \left[\bigsqcup^n_{i = 1}S^1, S^1\right] \simeq \mathbb{Z}^n$ because two maps are homotopic iff they are homotopic on every (path) connected component so it boils down to $[S^1, S^1] = \mathbb{Z}$ for $n$-times.
