Lemma 97.27.4. In Situation 97.27.1 the rule $F$ that sends a locally Noetherian scheme $V$ over $S$ to the set of triples $(Z, u', \hat x)$ satisfying the compatibility condition and which sends a a morphism $\varphi : V_2 \to V_1$ of locally Noetherian schemes over $S$ to the map

sending an element $(Z_1, u'_1, \hat x_1)$ of $F(V_1)$ to $(Z_2, u'_2, \hat x_2)$ in $F(V_2)$ given by

$Z_2 \subset V_2$ is the inverse image of $Z_1$ by $\varphi $,

$u'_2$ is the composition of $u'_1$ and $\varphi |_{V_2 \setminus Z_2} : V_2 \setminus Z_2 \to V_1 \setminus Z_1$,

$\hat x_2$ is the composition of $\hat x_1$ and $\varphi _{/Z_2} : V_{2, /Z_2} \to V_{1, /Z_1}$

is a contravariant functor.

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