Yet more explicitly, the wide pullback of a family of coterminal morphisms $f_i\colon A_i \to C$ is an object $P$ equipped with projection $p_i\colon P\to A_i$ such that $f_i p_i$ is independent of $i$, and which is universal with this property.

Binary wide pullbacks are the same as ordinary pullbacks, a.k.a. fiber products.

Of course, a wide pushout is a wide pullback in the opposite category.

The saturation of the class of wide pullbacks is the class of limits over categories $C$ whose fundamental groupoid$\Pi_1(C)$ is trivial.

On the other hand, together with a terminal object, wide pullbacks generate all limits:

Proposition

A category$C$ with all wide pullbacks and a terminal object$1$ is complete. If $C$ is complete and $F: C \to D$ preserves wide pullbacks and the terminal object, then it preserves all limits.

Proof

To build up arbitrary products $\prod_{i \in I} c_i$ in $C$, take the wide pullback of the family $c_i \to 1$. Then to build equalizers of diagrams $f, g: c \stackrel{\to}{\to} d$, construct the pullback of the diagram

$\array{
& & d \\
& & \downarrow \delta \\
c & \underset{\langle f, g \rangle}{\to} & d \times d
}$

From products and equalizers, we can get arbitrary limits.