Lemma 34.3.2 (020P)—The Stacks project

Lemma 34.3.2. Let $T$ be a scheme.

If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a Zariski covering of $T$.
If $\{ T_ i \to T\} _{i\in I}$ is a Zariski covering and for each $i$ we have a Zariski covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a Zariski covering.
If $\{ T_ i \to T\} _{i\in I}$ is a Zariski covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is a Zariski covering.

Proof. Omitted. $\square$

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