Lemma 79.4.2. Let $S$ be a scheme. Let $\mathcal{P}$ be a property as in Definition 79.4.1. Let

be a fibre square of presheaves on $(\mathit{Sch}/S)_{fppf}$. If $a$ is representable by algebraic spaces and has $\mathcal{P}$ so does $a'$.

Lemma 79.4.2. Let $S$ be a scheme. Let $\mathcal{P}$ be a property as in Definition 79.4.1. Let

\[ \xymatrix{ G' \times _ G F \ar[r] \ar[d]^{a'} & F \ar[d]^ a \\ G' \ar[r] & G } \]

be a fibre square of presheaves on $(\mathit{Sch}/S)_{fppf}$. If $a$ is representable by algebraic spaces and has $\mathcal{P}$ so does $a'$.

**Proof.**
Omitted. Hint: This is formal.
$\square$

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