multiplicativity of the singular homological degree on the sphere

Given the n-sphere \(S^n\) and continuous maps \(f,g: S^n \rightarrow S^n\), we get for the singular homological degree

\begin{align*} \mathrm{deg}(f \circ g) = \mathrm{deg}(f) \cdot \mathrm{deg}(g) \end{align*}

(n = 0) ?

By functorality of the singular homology functor, we get

\begin{align*} \mathrm{H}_{n}(f \circ g) = \mathrm{H}_n(f) \circ \mathrm{H}_n(g) \end{align*}

and by definition of the singular homological degree as endomorphism ring, we get for the ring-isomorphism

\begin{align*} \varphi: \mathrm{End}(\mathbb{Z}) \rightarrow \mathbb{Z} \end{align*}

the identity

\begin{align*} \mathrm{deg}(f \circ g) =& \varphi( \mathrm{H}_n(f) \circ \mathrm{H}_n(g)) \\ =& \varphi(\mathrm{H}_n(f)) \cdot \varphi(\mathrm{H}_n(g)) \\ =& \mathrm{deg}(f) \cdot \mathrm{deg}(g) \end{align*}