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Remark 4.14.4. We often write $\mathop{\mathrm{lim}}\nolimits _ i M_ i$, $\mathop{\mathrm{colim}}\nolimits _ i M_ i$, $\mathop{\mathrm{lim}}\nolimits _{i\in \mathcal{I}} M_ i$, or $\mathop{\mathrm{colim}}\nolimits _{i\in \mathcal{I}} M_ i$ instead of the versions indexed by $\mathcal{I}$. Using this notation, and using the description of limits and colimits of sets in Section 4.15 below, we can say the following. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram.

  1. The object $\mathop{\mathrm{lim}}\nolimits _ i M_ i$ if it exists satisfies the following property

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

    where the limit on the right takes place in the category of sets.

  2. The object $\mathop{\mathrm{colim}}\nolimits _ i M_ i$ if it exists satisfies the following property

    \[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(\mathop{\mathrm{colim}}\nolimits _ i M_ i, W) = \mathop{\mathrm{lim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) \]

    where on the right we have the limit over the opposite category with value in the category of sets.

By the Yoneda lemma (and its dual) this formula completely determines the limit, respectively the colimit.

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