singular homological degree invariant under conjugation

Given the n-sphere, a continuous map \(f: S^n \rightarrow S^n\) and an homeomorphism \(\varphi: S^n \rightarrow S^n\), it holds for the singular homological degree

\begin{align*} \mathrm{deg}(f) = \mathrm{deg}(\varphi \circ f \circ \varphi^{-1}) \end{align*}

corollary of

1. multiplicativity of the singular homological degree on the sphere
2. commutativity of \(\mathbb{Z}\)
3. singular homological degree of the identity on the sphere

as

\begin{align*} \mathrm{deg}(\varphi \circ f \circ \varphi^{-1}) =& \mathrm{deg}(\varphi) \cdot \mathrm{deg}(f) \cdot \mathrm{deg}(\varphi^{-1} \\ =& \mathrm{deg}(\varphi) \cdot \mathrm{deg}(\varphi^{-1}) \cdot \mathrm{deg}(f) \\ =& \mathrm{deg}(\varphi \circ \varphi^{-1}) \cdot \mathrm{deg}(f) \\ =& \mathrm{deg}(\mathrm{id}_{S^n}) \cdot \mathrm{deg}(f) \\ =& 1 \cdot \mathrm{deg}(f) \\ =& \mathrm{deg}(f) \end{align*}