Realize union of subsets of a given set as a colimit I am a little confused by one of Vakil's exercises, which reads 

Interpret the union of some subsets of a given set as a colimit. The objects of the category in question are the subsets of the given set.

Of course one could just take the diagram just consisting of the union $U$ itself (and the identity morphism). If the union is countable and $U_i \subset U_{i+1}$ for each $i$, then $U$ is the colimit of the diagram
$$
U_1 \to U_2 \to U_3 \to \cdots
$$
But what am I supposed to do with the case of an arbitrary index set $I$? I could also take some $i_0 \in I$ and consider the diagram $\bigcup\limits_{i\in I\setminus \{i_0\}} U_i \to U$.
$\textbf{Q}$: Does anyone have a clue what the author's intention was with this exercise?
 A: The power set $\mathcal{P}(X)$ of a set $X$ can be turned into a poset category: the objects of $\mathcal{P}(X)$ are the subsets of $X$, and the morphisms to be inclusions of subsets, i.e. there is a (unique) morphism $U \to V$ if and only if $U \subseteq V$. Identity comes from reflexivity of $\subseteq$, composition comes from transitivity of $\subseteq$, and the category axioms are trivially satisfied.
Moreover, an arbitrary set $I$ can be turned into a discrete category: the objects of $I$ are the elements of $I$, and there are only identity morphisms.
A collection of subsets $\{ U_i \subseteq X \mid i \in I \}$ can then be taken to be a diagram in the poset category $\mathcal{P}(X)$ indexed by the discrete category $I$. That is, we've defined a functor $F : I \to \mathcal{P}(X)$ by $F(i)=U_i$ for each $i \in I$. (Notice that the action of $F$ on morphisms is determined by the fact that the only morphisms in $I$ are identity morphisms.)
You can check that the colimit of $F$ is $\bigcup_{i \in I} U_i$. Moreover, the limit of $F$ is $\bigcap_{i \in I} U_i$.
