  Let $X$ be a smooth compact connected manifold. Let $G=\mbox{Diff}\, X$ be
the group of diffeomorphisms of $X$, equipped with the $C^\infty$-topology, and
let $H$ be the stabilizer of some point in $X$. Then the inclusion $H\to G$,
which is a morphism of two regular Fr\'echet--Lie groups, is an epimorphism in
the category of smooth Lie groups modelled on complete locally convex spaces.
At the same time, in the latter category, epimorphisms between finite
dimensional Lie groups have dense range. We also prove that if $G$ is a
Banach--Lie group and $H$ is a proper closed subgroup, the inclusion $H\to G$
is not an epimorphism in the category of Hausdorff groups.
