The Stacks project

Definition 4.14.1. A limit of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ in $\mathcal{C}$ together with morphisms $p_ i : \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \to M_ i$ such that

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

2. for any object $W$ in $\mathcal{C}$ and any family of morphisms $q_ i : W \to M_ i$ (indexed by $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $q_{i'} = M(\phi ) \circ q_ i$ there exists a unique morphism $q : W \to \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ such that $q_ i = p_ i \circ q$ for every object $i$ of $\mathcal{I}$.