The Stacks project

Remark 78.20.4. In future chapters we will use the ambiguous notation where instead of writing $\mathcal{S}_ X$ for the stack in sets associated to $X$ we simply write $X$. Using this notation the diagram of Lemma 78.20.3 becomes the familiar diagram

\[ \xymatrix{ R \ar[r]_ s \ar[d]_ t & U \ar[d]^\pi \\ U \ar[r]^-\pi & [U/R] } \]

In the following sections we will show that this diagram has many good properties. In particular we will show that it is a $2$-fibre product (Section 78.22) and that it is close to being a $2$-coequalizer of $s$ and $t$ (Section 78.23).

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