Alternating squarefree prime factors make mu undo an unrestricted divisor sum. The example is deliberately concrete; it is a test of the statement, not a substitute for it.
Start locally
Arithmetic functions \(f:\mathbf N\to\mathbf C\) form a commutative ring under Dirichlet convolution. Multiplicative functions are determined by their values on prime powers \(p^k\).
The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).
Compute before generalising
The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.
The global view
The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.
Edge conditions
Pointwise multiplication and Dirichlet convolution are different operations. Möbius inversion reverses convolution with the constant-one function, not ordinary multiplication.
I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.