Kock-Lawvere axiom for the generic local ring ($R^D\cong R\times R$)

For $D=\{d\in R:d^2=0\}$ in the generic local ring, every internal map $D\to R$ has a unique affine expression $d\mapsto a+bd$. At the stage $A$, such maps are elements of $A[\epsilon ]/({\epsilon }^2)$, which is freely $A\oplus A\epsilon$ as an $A$-module. This decomposition is natural under ring maps and proves the isomorphism. Testing only on ordinary fields loses the generic infinitesimal and would give an incorrect uniqueness argument.