The Stacks project

Lemma 4.17.6. Let $\mathcal{I}$ and $\mathcal{J}$ be a categories and denote $p : \mathcal{I} \times \mathcal{J} \to \mathcal{J}$ the projection. If $\mathcal{I}$ is connected, then for a diagram $M : \mathcal{J} \to \mathcal{C}$ the colimit $\mathop{\mathrm{colim}}\nolimits _\mathcal {J} M$ exists if and only if $\mathop{\mathrm{colim}}\nolimits _{\mathcal{I} \times \mathcal{J}} M \circ p$ exists and if so these colimits are equal.

Proof. This is a special case of Lemma 4.17.5. $\square$

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