This tag has label descent-lemma-cartesian-equivalent-descent-datum and it points to
The corresponding content:
Lemma 31.36.5. Let $f : X \to S$ be a morphism of schemes. The construction $$ \begin{matrix} \text{category of cartesian } \\ \text{schemes over } (X/S)_\bullet \end{matrix} \longrightarrow \begin{matrix} \text{ category of descent data} \\ \text{ relative to } X/S \end{matrix} $$ of Lemma 31.36.4 is an equivalence of categories.Proof. The functor from left to right is given in Lemma 31.36.4. Hence this is a special case of Lemma 31.36.2. $\square$
\begin{lemma}
\label{lemma-cartesian-equivalent-descent-datum}
Let $f : X \to S$ be a morphism of schemes. The construction
$$
\begin{matrix}
\text{category of cartesian } \\
\text{schemes over } (X/S)_\bullet
\end{matrix}
\longrightarrow
\begin{matrix}
\text{ category of descent data} \\
\text{ relative to } X/S
\end{matrix}
$$
of Lemma \ref{lemma-cartesian-over}
is an equivalence of categories.
\end{lemma}
\begin{proof}
The functor from left to right is given in
Lemma \ref{lemma-cartesian-over}.
Hence this is a special case of
Lemma \ref{lemma-characterize-cartesian-schemes}.
\end{proof}
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