Let \(G\) be a connected reductive group. Previously, it was shown that for any \(G\)-variety \(X\) one can define the dual group \(G^\vee_X\) which 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 unique homomorphism with finite kernel \(G^\vee_Y\to G^\vee_X\) which is compatible with the homomorphisms to \(G^\vee\).
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