Lemma 42.61.3. Let $k$ be a field. Let $X$ be a scheme locally of finite type over $k$. Let $c \in A^ p(X \to \mathop{\mathrm{Spec}}(k))$. Let $Y \to Z$ be a morphism of schemes locally of finite type over $k$. Let $c' \in A^ q(Y \to Z)$. Then $c \circ c' = c' \circ c$ in $A^{p + q}(X \times _ k Y \to X \times _ k Z)$.

Proof. In the proof of Lemma 42.61.2 we have seen that $c$ is given by a combination of proper pushforward, multiplying by integers over connected components, and flat pullback. Since $c'$ commutes with each of these operations by definition of bivariant classes, we conclude. Some details omitted. $\square$

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