Definition 4.26.1. Let $\mathcal{C}$ be a big^{1} category. An object $X$ of $\mathcal{C}$ is called a *categorically compact* if we have

\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, \mathop{\mathrm{colim}}\nolimits _ i M_ i) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, M_ i) \]

for every filtered diagram $M : \mathcal{I} \to \mathcal{C}$ such that $\mathop{\mathrm{colim}}\nolimits _ i M_ i$ exists.

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