What kind of object is "the product of all objects of a category"? Let us denote the set of all objects of a small complete category by $C^{\bullet}$. My question is concerned with the limit of the diagram $$C^{\bullet} \longrightarrow C$$ which sends every morphism of $C^{\bullet}$ which they all happen to be identities, to the identities. What kind of object is the limit (or colimit for that matter) of the above diagram. For example the category of finite sets doesn't have the product of all of its objects. Perhaps I must look for more peculiar categories than **FinSet** for meeting such a beast. Thanks.

Here is a relevant result showing that such categories are probably rare. Freyd showed that if a small category has all small limits, then it must be a preorder. So we can more or less reduce to the case that $C$ is a poset, in which case the product of all of the objects is a smallest element (if it exists).