Lemma 4.16.3 (04AR)—The Stacks project

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Lemma 4.16.3. Let $\mathcal{C}$ be a category. Let $X$ be an object of $\mathcal{C}$ . Let $M : \mathcal{I} \to X/\mathcal{C}$ be a diagram in the category of objects under $X$ . If the index category $\mathcal{I}$ is connected and the colimit of $M$ exists in $X/\mathcal{C}$ , then the colimit of the composition $\mathcal{I} \to X/\mathcal{C} \to \mathcal{C}$ exists and is the same.

Proof. Omitted. Hint: This lemma is dual to Lemma 4.16.2. $\square$

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