Commutativity of centralizers in a coproduct of a free algebra and a polynomial algebra

We show that the centralizer of a nonscalar element in the coproduct $k⟨X⟩\ast k[Y]$ of a free associative algebra and a polynomial algebra over a given field is commutative. For $k⟨X⟩$ this is part of Bergman’s centralizer theorem. Our proof relies on a reduction given in Bergman’s proof and is of combinatorial nature, employing a strict order structure of the coproduct monoid.