Lemma 4.17.4 (002R)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.17: Cofinal and initial categories
Lemma 4.17.4 (cite)

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Lemma 4.17.4. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of categories. Assume $\mathcal{I}$ is initial in $\mathcal{J}$ . Then for every diagram $M : \mathcal{J} \to \mathcal{C}$ we have a canonical isomorphism

$\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \circ H = \mathop{\mathrm{lim}}\nolimits _\mathcal {J} M$

if either side exists.

Proof. Omitted. $\square$

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