The Stacks project

Lemma 13.14.14. Assumptions and notation as in Situation 13.14.1.

  1. If for every object $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ there exists an arrow $s : X \to X'$ in $S$ such that $X'$ computes $RF$, then $RF$ is everywhere defined.

  2. If for every object $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ there exists an arrow $s : X' \to X$ in $S$ such that $X'$ computes $LF$, then $LF$ is everywhere defined.

Proof. This is clear from the definitions. $\square$

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