Cofiltered Category

A category is finitely cofiltered if every finite diagram has a cone.

In more elementary terms, a category $\mathcal{C}$ is cofiltered if:

• It is merely inhabited.
• For any $X,Y:\mathcal{C}$, there merely exists an $A:\mathcal{C}$ with $A\to X$, $A\to Y$.
• For any two parallel morphisms $f,g:X\to Y$, there exists an object $A$ and map $e:A\to X$ such that $fe=ge$.

In other words, we have mere existence of a diagram satisfying the requirements of a Terminal Object, Cartesian Product, and Equaliser, resp.