Finitely complete category

Remarks

The following are equivalent for a category $\mathcal{C}$:

1. $\mathcal{C}$ is finitely complete.
2. $\mathcal{C}$ has a terminal object, binary products and equalizers.
3. $\mathcal{C}$ has a terminal object and pullbacks.

Example: the category of finite sets is finitely complete. Its terminal object is a one-point set, and the pullback of $f \colon X \to Z$ and $g \colon Y \to Z$ is the finite set $\set{(x, y) : f(x) = g(y)}$.

Non-example: the totally ordered set $(\Z, \le)$, seen as a category, has all pullbacks but no terminal object. In a preorder the pullback of $x \to z \leftarrow y$ is $\min(x, y)$, and a terminal object would be a greatest element.