Lemma 42.59.7. Let $(S, \delta )$ be as in Situation 42.7.1. Consider a commutative diagram

of schemes locally of finite type over $S$ with both square cartesian. Assume $f : X \to Y$ is a local complete intersection morphism such that the gysin map exists for $f$. Let $c \in A^*(Y'' \to Y')$. Denote $res(f^!) \in A^*(X' \to Y')$ the restriction of $f^!$ to $Y'$ (Remark 42.33.5). Then $c$ and $res(f^!)$ commute (Remark 42.33.6).

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