= Commutation of limits and colimits
{wiki=Limit_(category_theory)#Commutation_of_limits_and_colimits}
Limits of shape $\mathcal I$ commute with colimits of shape $\mathcal J$ when, for every $D:\mathcal I\times\mathcal J\to\mathcal C$, the canonical comparison
$$
\operatorname*{colim}_{j\in\mathcal J}\operatorname*{lim}_{i\in\mathcal I}D(i,j)
\longrightarrow
\operatorname*{lim}_{i\in\mathcal I}\operatorname*{colim}_{j\in\mathcal J}D(i,j)
$$
is an isomorphism whenever both sides exist.
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