In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism.

As pointed out below, the functor does not preserve all epimorphisms in the category of Lie groups, but only those which are also epimorphisms (surjective) in the ambient topos of sets.

Does this continue to hold in synthetic differential geometry? In this case, it seems that the appropriate definition of a "surjective" Lie group homomorphism is a Lie group homomorphism which is an epi within the ambient smooth topos.

To start, suppose we have a "surjective" Lie group homomorphism: $\phi: G \to H$. That is, for maps $\psi_1: H \to M$ and $\psi_2: H \to M$, if $\psi_1 \circ \phi = \psi_2 \circ \phi$ then $\psi_1 = \psi_2$.

Now if $f_1 : \mathfrak{h} \to N$ and $f_2 : \mathfrak{h} \to N$ are any maps, we suppose that $f_1 \circ \phi_* = f_2 \circ \phi_*$. Then for any Lie algebra element $(X: D \to G) \in \frak{g}$, we have $$f_1 (\phi \circ X) = f_2 (\phi \circ X).$$ But I am not sure how to conclude that $f_1 = f_2$. Indeed, as far as I know covariant representable functors do not in general preserve epimorphisms.

Sorry if this is obvious; I like to read up on SDG as a hobby, but I am not an expert in category theory or topos theory.