If all coproducts and coequalizers exist, all colimits exist.

Lemma 4.14.12. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. Write $I = \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and $A = \text{Arrows}(\mathcal{I})$. Denote $s, t : A \to I$ the source and target maps. Suppose that $\coprod _{i \in I} M_ i$ and $\coprod _{a \in A} M_{s(a)}$ exist. Suppose that the coequalizer of

$\xymatrix{ \coprod _{a \in A} M_{s(a)} \ar@<1ex>[r]^\phi \ar@<-1ex>[r]_\psi & \coprod _{i \in I} M_ i }$

exists, where the morphisms are determined by their components as follows: The component $M_{s(a)}$ maps via $\psi$ to the component $M_{t(a)}$ via the morphism $a$. The component $M_{s(a)}$ maps via $\phi$ to the component $M_{s(a)}$ by the identity morphism. Then this coequalizer is the colimit of the diagram.

Proof. Omitted. $\square$

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