Anders Kock mentions many axioms in his book about synthetic differential geometry. It seems that, what nlab calls: "the Kock-Lawvere axiom", is called axiom $1^W_k$ in the book. This axiom states that for a Weil algebra $W$ over $k$ we have that $R \otimes W$ is isomorphic to $R^{\text{Spec}_R(W)}$.
This makes a decent amount of sense to me, however in this book axiom $1^W_k$ is actually a consequence of axiom $2^k$, which he calls: "The comprehensive axiom". This axiom states that for finitely presented $k$-algebras $B$, and any $R$-algebra $C$, the canonical map $$\text{hom}_{R\text{-Alg}}(R^{\text{Spec}_R(B)},C) \to \text{Spec}_C(B)$$ is an isomorphism.
I don't quite understand this axiom and I especially don't get why one would assume this, as it is not necessarily about infinitesimals.
So why is this axiom assumed? Is this axiom maybe just a standard result in algebraic geometry, and if so which?