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. 
 A: 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.
A: 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). 
A: A simple example: let $X$ be a set, $P_X$ be the preorder of its subsets, ordered by inclusion. Then the product of all objects in $P_X$ will be $\varnothing$(empty subset of $X$), and coproduct of all objects in $P_X$(which is colimit of your diagram) will be $X$(which is also subset of $X$).
