  Let $G$ be a connected reductive group. In a previous paper,
arxiv:1702.08264, is was shown that the dual group $G^\vee_X$ attached to a
$G$-variety $X$ admits a natural homomorphism with finite kernel to the
Langlands dual group $G^\vee$ of $G$. Here, we prove that the dual group is
functorial in the following sense: if there is a dominant $G$-morphism $X\to Y$
or an injective $G$-morphism $Y\to X$ then there is a canonical homomorphism
$G^\vee_Y\to G^\vee_X$ which is compatible with the homomorphisms to $G^\vee$.
