Functoriality properties of the dual group

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 $G^\vee$.