The Stacks project

  • For any countable1 diagram category $\mathcal{I}$ and any functor $F : \mathcal{I} \to \mathit{Sch}_\alpha $, the limit $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} F$ exists in $\mathit{Sch}_\alpha $ if and only if it exists in $\mathit{Sch}$ and moreover, in this case, the natural morphism between them is an isomorphism.

[1] Both the set of objects and the morphism sets are countable. In fact you can prove the lemma with $\aleph _0$ replaced by any cardinal whatsoever in (3) and (4).

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