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 \\(+\\).