Lemma 42.47.3. In Lemma 42.47.1 let $X' \to X$ be a morphism which is locally of finite type. Denote $X' = X'_1 \cup X'_2$ and $E'_2 \in D(\mathcal{O}_{X'_2})$ the pullbacks to $X'$. Then the class $P'_ p(E_2')$, resp. $c'_ p(E_2')$ in $A^ p(X_2' \to X')$ constructed in Lemma 42.47.1 using $X' = X'_1 \cup X'_2$ and $E_2'$ is the restriction (Remark 42.33.5) of the class $P'_ p(E_2)$, resp. $c'_ p(E_2)$ in $A^ p(X_2 \to X)$.

**Proof.**
Immediate from the characterization of these classes in Lemma 42.47.1.
$\square$

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