= Transported addition on the multiplicative natural-number monoid
{title2=$a\oplus b=\phi^{-1}(\phi(a)+\phi(b))$}
For a multiplicative <monoid automorphism> $\phi:\mathbb N\to\mathbb N$ fixing $0$, define $a\oplus b=\phi^{-1}(\phi(a)+\phi(b))$. This transports a <semiring> structure to the fixed underlying multiplication, hence a <commutative-monoid enrichment> on its one-object category. Swapping prime $2$ with an <odd prime> $p$ gives $1\oplus1=p$. Infinitely many such primes therefore give infinitely many distinct additions, although the transported enriched categories are isomorphic.
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