The Stacks project

Proof. It is clear that (3) implies (2). Assume (2). Then $\text{id}_ X \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, X)$ comes from a morphism $X \to M_ i$ for some $i$, i.e., $X \to M_ i \to X$ is the identity. Then both maps

are bijective for all $W$ where the first one is induced by the morphism $X \to M_ i$ we found above, and the composition is the identity. This means that the composition

is the identity too. Setting $W = M_ j$ and starting with $\text{id}_{M_ j}$ in the colimit, we see that $M_ j \to X \to M_ i \to M_ k$ is equal to $M_ j \to M_ k$ for some $k$ large enough. This proves (1) holds.

Assume (4). Let $k$ be an object of $\mathcal{I}$. Setting $W = M_ k$ we deduce there exists a unique morphism $M_ k \to X$ such that there exists a $j$ and morphisms $k \to j$ and $i \to j$ in $\mathcal{I}$ such that $M_ k \to X \to M_ i \to M_ j$ is equal to $M_ k \to M_ j$. The uniqueness guarantees that we obtain a cocone $(X, \{ M_ k \to X\} )$. In this way we see that (4) implies (2); some details omitted.

We omit the proof that (1) implies the other conditions. $\square$