The Stacks project

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

\[ \xymatrix{ & W \ar[dl]_{q_ i} \ar[dr]^{q_{i'}} \\ M_ i \ar[rr]^{M(\phi )} & & M_{i'} } \]

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

\[ \xymatrix{ M_ i \ar[rr]^{M(\phi )} \ar[dr]_{t_ i} & & M_{i'} \ar[dl]^{t_{i'}} \\ & W } \]

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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