Lemma 40.11.3. Notation and assumptions as in Situation 40.11.1. Let $Z \subset R_2$ be the reduced closed subscheme (see Schemes, Definition 26.12.5) whose underlying topological space is the closure of the image of $a : R_1 \to R_2$. Then $c_2(Z \times _{s_2, U, t_2} Z) \subset Z$ set theoretically.

**Proof.**
Consider the commutative diagram

\[ \xymatrix{ R_1 \times _{s_1, U, t_1} R_1 \ar[r] \ar[d] & R_1 \ar[d] \\ R_2 \times _{s_2, U, t_2} R_2 \ar[r] & R_2 } \]

By Varieties, Lemma 33.24.2 the closure of the image of the left vertical arrow is (set theoretically) $Z \times _{s_2, U, t_2} Z$. Hence the result follows. $\square$

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