Lemma 76.17.4. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $X \subset X'$ and $Y \subset Y'$ be first order thickenings over $B$. Assume given a morphism $a : X \to Y$ and a map $A : a^*\mathcal{C}_{Y/Y'} \to \mathcal{C}_{X/X'}$ of $\mathcal{O}_ X$-modules. For an object $U'$ of $(X')_{spaces, {\acute{e}tale}}$ with $U = X \times _{X'} U'$ consider morphisms $a' : U' \to Y'$ such that

$a'$ is a morphism over $B$,

$a'|_ U = a|_ U$, and

the induced map $a^*\mathcal{C}_{Y/Y'}|_ U \to \mathcal{C}_{X/X'}|_ U$ is the restriction of $A$ to $U$.

Then the rule

defines a sheaf of sets on $(X')_{spaces, {\acute{e}tale}}$.

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