TITLE: What kind of object is "the product of all objects of a category"?
QUESTION [5 upvotes]: 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.

REPLY [7 votes]: I would dispute the above answers that we can (mostly) restrict our attention to preorders; for example, the category $\mathsf{FinSet}$ does have a product of all objects, and it is simply the empty set with empty projections. Similarly, $\mathsf{Set}$ and $\mathsf{Top}$ contain products of all objects, and many more similar examples can be given.
Edit: Ah, aws posted this a minute before me; I did not see the comment in time.