= Kock-Lawvere axiom for the generic local ring
{c}
{title2=$R^D\cong R\times R$}
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 $A[\varepsilon]/(\varepsilon^2)$, which is freely $A\oplus A\varepsilon$ 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.
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