I have a question about Chern class of symplectic reduction.

Let $(M,\omega)$ be a smooth compact symplectic manifold with a Hamiltonian circle action. Let $H : M \rightarrow \mathbb{R}$ be the corresponding moment map. When $0$ is a regular value of $H$, how can I prove that a Kirwan map $\kappa : H^*_{S^1}(M) \rightarrow H^*(H^{-1}(0)/S^1)$ maps the equivariant first Chern class of $M$ to the first Chern class of $H^{-1}(0)/S^1$?

I saw this argument in the paper "Floer homology on the extended moduli space" by Manolescu and Woodward (Lemma 4.4), but the precise proof is not involved. Perhaps (I guess) the proof of the statement seems to be an exercise, but I don't know what I missed.

Thank you in advance.