Classically the Lie functor maps a Lie group homomorphism to a Lie algebra homomorphism. But in Proposition 15 on page 249 in Basic Concepts of Synthetic Differential Geometry, Lavendhomme states that if $\alpha: G \to \text{Diff}(M)$ is a left action (a group homomorphism) then $T_e \alpha: \frak{g} \to \frak{X}$$(M)$ is an antihomorphism of Lie algebras.
It seems that this is not a typo, and he proves that this is indeed the case. Is there something I'm missing here?
If $G$ is a Lie group (a microlinear group), then the Lie algebra consists of all $X: D \to G$ such that $X(0)=e$. Here $D = \{ d \in \mathbb{R} : d^2 =0 \}$ is the "walking tangent vector." The Lie bracket of $X, Y \in \frak{g}$ is then defined to be the unique element $[X, Y] \in \frak{g}$ such that $$[X, Y](d_1 \cdot d_2) = Y(-d_2) \cdot X(-d_1) \cdot Y(d_2) \cdot X(d_1)$$ for any $d_1, d_2 \in D$. If $\text{Diff}(M)$ is microlinear, then the tangent space at $\text{id}_M$ is $\frak{X}$$(M)$, so this defines the Lie bracket on vector fields as well. This appears to be equivalent to the usual definition of the Lie bracket in terms of flows.