Thanks Jesus, this is what I've missed - exclusion of 0 (\$$f\$$) when searching for multiplicative inverses.

Indeed, we have 2 categories, which are separately mapped to \$$(\mathbb{Z}/2, +)\$$ and \$$(\mathbb{Z}/2, \cdot)\$$, so \$$f \cong 0\$$ under \$$\cdot\$$, but \$$f \cong 1\$$ under \$$+\$$.