Proof. Suppose we have a direct product of $\{C_i\}_{i\in I}$ for some ( ) set $I$ . Consider $I$ as a category whose arrows are only the identity arrows. Then we can define a functor $G$ by $G(i)=C_i$ . It is then clear that the universal property of an inverse limit is equivalent to the universal property defining a categorical direct product.

Reversing the arrows, it is also clear that the categorical direct sum is an example of a direct limit.

These results are of interest when one is looking to prove exactness of sums and products in a category: often it is easier to address exactness of direct and inverse limits, and the result then applies to many other constructions as well.