Remark 77.19.2. A variant of the construction above would have been to sheafify the functor

where now $\sim _ X \subset U(X) \times U(X)$ is the equivalence relation generated by the image of $j : R(X) \to U(X) \times U(X)$. Here of course $U(X) = \mathop{\mathrm{Mor}}\nolimits _ B(X, U)$ and $R(X) = \mathop{\mathrm{Mor}}\nolimits _ B(X, R)$. In fact, the result would have been the same, via the identifications of (insert future reference in Topologies of Spaces here).

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