I know that the $\mu$ totient function have this two important properties:
The first one is that, supposing that $f$ is a multiplicative arithmetic function, I have that $g=f\star u$ if and only if $f=\mu \star g$, where $\star$ is the convolution of Dirichlet, $u(n)=1, \forall n\in \mathbb{N}^{\times}$, and $\mu$ is the Möbius function.
The second one is that if I have the function $I$ such that $I(n)=1$ if $n=1$ and $I(n)=0$ if $n\neq 1$, then I have that $\mu \star u=I$, being $u$ defined as the last paragraph.
These two properties have easy proofs, but I don't know how to prove that $\mu$ is the ONLY ONE function that satisfies those two properties, i.e. I have to prove the uniqueness from the $\mu$ function for the properties.