# The Stacks Project

## Tag: 06PM

This tag has label stacks-properties-lemma-descent-surjective and it points to

The corresponding content:

Lemma 68.5.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Let $\mathcal{Y}' \to \mathcal{Y}$ be a surjective morphism of algebraic stacks. If the base change $f' : \mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \to \mathcal{Y}'$ of $f$ is surjective, then $f$ is surjective.

Proof. Immediate from Lemma 68.4.3. $\square$

\begin{lemma}
\label{lemma-descent-surjective}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
Let $\mathcal{Y}' \to \mathcal{Y}$ be a surjective morphism of algebraic
stacks. If the base change $f' : \mathcal{Y}' \times_\mathcal{Y} \mathcal{X} \to \mathcal{Y}'$ of $f$ is surjective, then $f$ is surjective.
\end{lemma}

\begin{proof}
Immediate from
Lemma \ref{lemma-points-cartesian}.
\end{proof}


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