Complete categories are cocomplete?

Question

I've read in a paper and on wikipedia that any (small) category is complete if and only if it is cocomplete. Now obviously if one shows that complete$\implies$cocomplete, then it's easy to conclude from there, but I have no idea why that would be true. Could anyone care to explain it to me ?

Answer 1

This is not true in general for a given category $C$. On the other hand, note that $\mathbf{Set}^{C^{op}}$, $is$ complete and cocomplete, and that the Yoneda Lemma provides an embedding $\ y \colon C \rightarrow \mathbf{Set}^{C^{op}}. $ That is, $C$ can be "enlarged" to a category that is complete and cocomplete.

Answer 2

The point is that a small complete category is a complete preorder. See nlab article.

Answer 3

The partial ordered class of sets has all suprema, but not all infima, since it lacks a largest element. Hence, its dual category is complete and not cocomplete.