On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$ $$\require{AMScd} \begin{CD} M^D @>{f^D}>> N^D\\ @V{M^d}VV @VV{N^d}V\\ M^\mathbf{1} @>>{f^\mathbf{1}}> N^\mathbf{1}. \end{CD}$$ Concretely this means the tangent space $M^D=TM$ is isomorphic to set of pairs $(t,x)\in N^D\times M$ satisfying $t(d)=f(x)$. This is not the same as just taking tangents whose basepoint $t(0)=f(x)$, so how is this definition equivalent to the one saying the induced map on tangent spaces is an iso?
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You have omitted a crucial part of the sentence:
... in the sense of §16.
The point is small objects admit canonical basepoints via the augmentation of the Weil algebras defining them. See this MSE question for the definition of $\operatorname{Spec}_R(\pi)$ for the augmentation $\pi:W\rightarrow R$. The author's definition only asks for pullbacks involving these canonical base points.
For the dual numbers, this notion does pick zero as a basepoint, since dual numbers can be identified with $R^2$ with-dual-number-multiplication, making the projection of $\bar x=(0,x)$ clearly zero.
