Remark 4.14.5. Let M : \mathcal{I} \to \mathcal{C} be a diagram. In this setting a cone for M is given by an object W and a family of morphisms q_ i : W \to M_ i, i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}) such that for all morphisms \phi : i \to i' of \mathcal{I} the diagram
is commutative. The collection of cones forms a category with an obvious notion of morphisms. Clearly, the limit of M, if it exists, is a final object in the category of cones. Dually, a cocone for M is given by an object W and a family of morphisms t_ i : M_ i \to W such that for all morphisms \phi : i \to i' in \mathcal{I} the diagram
commutes. The collection of cocones forms a category with an obvious notion of morphisms. Similarly to the above the colimit of M exists if and only if the category of cocones has an initial object.
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