No. The axiom is restricting the allowable maps $R^\Delta$ (and so also $R^R$) in the category.

@user10354138 I'm beginner and I can't understand what you mean. In the proof in the book there are no such terms. Please, give more details.

If you actually want to do synthetic differential geometry, you should at least have a good grasp of category theory and topos theory. I don't think it is appropriate to do a semester (or two) course on category theory and toposes here on math.SE.

@user10354138 I meant that Kock-Lawvere axiom help us to prove that there are nilsquare infinitesimals.

Wha-ah? It basically says all such mappings $\Delta \to R$ are linear. It doesn't prove that $\Delta$ is non-empty.

@BrianTung math.stackexchange.com/questions/4305853/…

Yes, and...so? That proof just points out that $\Delta$ can't only contain $0$, because then there isn't a unique $b$. What does this have to do with finding the infinitesimals? I guess I don't understand exactly what you're asking.

@BrianTung I said "finding set of infinitesimals". I mean that this set can't only contain $0$.

Yes, that's right. What are you asking then? Are you having trouble understanding the proof?

@BrianTung No. But do we check after this proof that $\varepsilon$ satisfy condition $\varepsilon^2=0$?

I've edited your question to use mathjax (which is searchable) rather than an image (which isn't) so that other users will have an easier time finding this question. In the future you should do the same ^_^

The proof is non-constructive; that is, it doesn't identify what other $\varepsilon \not= 0$ there must be in $\Delta$. So how can you check whether $\varepsilon^2 = 0$? (Other than it must be by virtue of being in $\Delta$.) ¶ I second @user10354138's comment. Are you sure you have a firm command of category theory and topos theory?

P.S. I would put that link in your question above, by the way, and mention it as part of your context. Your question is still somewhat confusing with that link, but without it, it's pretty unmotivated.

@BrianTung I can check that $\varepsilon^2=0$. This is given in the book: "So this $\Delta$ contains more points than only $0$. There is very elegant geometric illustration of $\Delta$ given by Joyal. It says that if you consider the unit circle in $R^2$ centered at point $(0,1)$, then the collection of points where it is tangent to the x-axis is precisely $\Delta$... The points at which the given circle is tangent to the x-axis satisfy $x^2=0$, which are precisely the infinitesimals on $R$".

That's an illustration, not a verification. May I ask what your mathematics background is—what courses you've taken, etc?

@BrianTung I only read books and articles about SDG and SIA. I'm at the beginnning. What courses do you recommend?

I don't mean just related to synthetic differential geometry. What mathematics courses have you taken in general? Linear algebra, single and multivariable calculus, real and complex analysis, geometry, number theory, differential equations, integration theory, probability theory, topology, category theory, you name it? ¶ I sense gaps in your understanding that suggest an incomplete background. Only you can say for sure what you know, but you must be scrupulously honest with yourself. I don't wish to quell your enthusiasm, but are you sure you understand everything as well as you think you do?

@BrianTung I have taken linear algebra, single and multivariable calculus, real and complex analysis, geometry, differential equations, integration theory, probability theory. I haven't taken number theory, topology and category theory. I sense gaps in my understanding too. I decided to self-educate and I have enthusiasm but some things are difficult for me. After you said above: "Are you having trouble understanding the proof?" I said "no", but now I understand that I have trouble. Can you explain something to me? Thanks.

I'm afraid I probably can't. In the first place, I don't have the time to do that, but more so, I don't know the prerequisites for this field in anything like the necessary comprehensiveness to explain the situation more broadly. You might try reaching out in one of the chats here to see if you can find someone who'd be willing to help you, but this thread is beginning to venture outside the Q&A realm that is the mainstream of this site. Sorry!

@BrianTung, You wrote: "The proof is non-constructive; that is, it doesn't identify what other ε≠0 there must be in Δ." This is a misconception. There are no provably nonzero elements in $\Delta$, so you can't write "$\varepsilon\not=0$."