# The Stacks Project

## Tag 08X3

Lemma 34.4.18. The diagram $$\tag{34.4.18.1} \xymatrix@C=8pc{ S_1 \ar[r]^{\delta^1_1} & S_2 \ar@<1ex>[r]^{\delta^2_2} \ar@<-1ex>[r]_{\delta^2_1} & S_3 }$$ is a split equalizer.

Proof. In Lemma 34.4.16, take $(M, \theta) = f^*(S)$. $\square$

The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 1161–1173 (see updates for more information).

\begin{lemma}
\label{lemma-equalizer-S}
The diagram

\label{equation-equalizer-S}
\xymatrix@C=8pc{
S_1 \ar[r]^{\delta^1_1} &
S_2 \ar@<1ex>[r]^{\delta^2_2} \ar@<-1ex>[r]_{\delta^2_1} &
S_3
}

is a split equalizer.
\end{lemma}

\begin{proof}
In Lemma \ref{lemma-equalizer-M}, take $(M, \theta) = f^*(S)$.
\end{proof}

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