Definition 83.8.1. Let $S$ be a scheme and $B$ an algebraic space over $S$. Let $j : R \to U \times _ B U$ be a pre-relation. Let $\phi : U \to X$ be an $R$-invariant morphism. Denote $\phi ' = \phi \circ s = \phi \circ t : R \to X$.
We denote $(\phi _*\mathcal{O}_ U)^ R$ the $\mathcal{O}_ X$-sub-algebra of $\phi _*\mathcal{O}_ U$ which is the equalizer of the two maps
\[ \xymatrix{ \phi _*\mathcal{O}_ U \ar@<1ex>[rr]^{\phi _*s^\sharp } \ar@<-1ex>[rr]_{\phi _*t^\sharp } & & \phi '_*\mathcal{O}_ R } \]on $X_{\acute{e}tale}$. We sometimes call this the sheaf of $R$-invariant functions on $X$.
We say the functions on $X$ are the $R$-invariant functions on $U$ if the natural map $\mathcal{O}_ X \to (\phi _*\mathcal{O}_ U)^ R$ is an isomorphism.
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