Implication Details

Conclusions: cofiltered limits, pullbacks

This is an equivalence.

Reason: To prove $\Leftarrow$ , a wide pullback can be constructed as a cofiltered limit of finite pullbacks, and finite pullbacks can be reduced to binary pullbacks (the empty-indexed pullback always exists). Conversely, assume that wide pullbacks exist in $\mathcal{C}$ . For every object $A$ then the slice category $\mathcal{C} / A$ has wide pullbacks and a terminal object, hence is complete. Since a cofiltered limit can be finally reduced to such a slice, we are done.