Kock defines, after (16.2), the canonical base point of a small object $\operatorname{Spec}_R(W)$ to be $$\mathbf 1\overset{\operatorname{Spec}_R (\pi)}{\longrightarrow}\operatorname{Spec}_RW$$where $\pi:W\rightarrow R$ is the augmentation map of the Weil algebra $W$, and the $\mathbf 1$ is just $\operatorname{Spec}_RR$. Thing is, I don't understand what $\operatorname{Spec}_R$ does to arrows as a functor, since there are no prime ideals to take inverse images of.
So, what does $\operatorname{Spec}_R$ do as a functor, and how does it follow that in the case of the dual numbers, it chooses zero?
See Lavendhomme's Basic Concepts of Synthetic Differential Geometry, chapter 2, section 2.1.1, after definition 1.
Given $f:W_1\rightarrow W_2$, choose finite presentations and define $\operatorname{Spec}_R(f)(b_1,\dots ,b_{r^\prime})=(a_1,\dots ,a_r)$ by $a_i=f(x_i)(b_1,\dots ,b_{r^\prime})$.