In synthetic differential geometry, an object $M$ verifies the *Wraith axiom* if for all functions $\tau:D\times D\to M$ which are constant on the axes $\tau(d,0)=\tau(0,d)=\tau(0,0)$ for all $d\in D$, there's a unique factorization through the multiplication map, i.e there's a unique function $t:D\to M$ such that $t(d_1\cdot d_2)=\tau(d_1,d_2)$.

What is the geometric/physical intuition behind this axiom? What is the analogue in the category of smooth manifolds?

**Update.** Following the answers I think I should add some motivation. Lavendhomme's book defines the commutator of vector fields as $\tau(d_1,d_2)=Y_{-d_2}X_{-d_1}Y_{d_2}X_{d_1}$. Because this is constant on axes, it factors through the multiplication map to give a vector field $t$ charaterized by $t(d_1d_2)=\tau(d_1,d_2)$. I understand $t$ is desirable since it is a vector field, but I don't know how to geometrically interpret its characterizing property. For instance, why not consider the vector field $\tau(d_1,d_1)$ given by precomposing the diagonal? This motivated my question.