Lemma 4.14.10. Let $\mathcal{I}$, $\mathcal{J}$ be index categories. Let $M : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$ be a functor. We have

$\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _ j M_{i, j} = \mathop{\mathrm{colim}}\nolimits _{i, j} M_{i, j} = \mathop{\mathrm{colim}}\nolimits _ j \mathop{\mathrm{colim}}\nolimits _ i M_{i, j}$

provided all the indicated colimits exist. Similar for limits.

Proof. Omitted. $\square$

There are also:

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