Complete category

Remarks

Dually, $\mathcal{C}$ is cocomplete if every small diagram has a colimit. A category is complete if and only if it has all small products and all equalizers.

Example: the category of sets is complete. The limit of $F \colon J \to \mathbf{Set}$ is the set of families $(x_j)_{j \in J}$ with $x_j \in F(j)$ and $F(u)(x_j) = x_k$ for every morphism $u \colon j \to k$ of $J$.

Non-example: the category of finite sets is not complete. Suppose $P$ were a product of countably many copies of $\set{0, 1}$ there. Maps from a one-point set to $P$ would match the elements of $P$ with all sequences in $\set{0, 1}$, which form an uncountable set.

A small category that is complete is a preorder, by a theorem of Freyd. So completeness is a property of large categories.