The Stacks project

Definition 4.14.2. A colimit of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M$ in $\mathcal{C}$ together with morphisms $s_ i : M_ i \to \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M$ such that

1. for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $s_ i = s_{i'} \circ M(\phi )$, and

2. for any object $W$ in $\mathcal{C}$ and any family of morphisms $t_ i : M_ i \to W$ (indexed by $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $t_ i = t_{i'} \circ M(\phi )$ there exists a unique morphism $t : \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \to W$ such that $t_ i = t \circ s_ i$ for every object $i$ of $\mathcal{I}$.