The Stacks project

Proof. Let $L \to X$ be an object representing the limit in $\mathcal{C}/X$. Consider the functor

Let $(\varphi _ i)$ be an element of the set on the right. Since each $M_ i$ comes equipped with a morphism $s_ i : M_ i \to X$ we get morphisms $f_ i = s_ i \circ \varphi _ i : W \to X$. But as $\mathcal{I}$ is connected we see that all $f_ i$ are equal. Since $\mathcal{I}$ is nonempty there is at least one $f_ i$. Hence this common value $W \to X$ defines the structure of an object of $W$ in $\mathcal{C}/X$ and $(\varphi _ i)$ defines an element of $\mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}/X}(W, M_ i)$. Thus we obtain a unique morphism $\phi : W \to L$ such that $\varphi _ i$ is the composition of $\phi$ with $L \to M_ i$ as desired. $\square$