$ \varphi \times 1=id $
$ μ \times id=\varphi $
$ μ \times 1=\varepsilon $
$ \varepsilon \times1 =1 $
$ \sum _{i=1}^{n}\sum_{d|i}{d\cdot\varphi(d)}=\sum _ {i=1}^{n}\sum _{d|i}(\frac{i}{d})\cdot \varphi(\frac{i}{d})=\sum_{i=1}^{n}\sum_{\frac{i}{d}=1} ^{\lfloor\frac{n}{i}\rfloor} (\frac{i}{d})\cdot \varphi(\frac{i}{d}) =\sum_{i=1}^{n}\sum_{d=1}^{\lfloor\frac{n}{i}\rfloor}{d\cdot\varphi(d)} $