Warning 184.108.40.206. The conclusion of Exercise 220.127.116.11 does not necessarily hold if the map $i$ is not injective. For example, let $G$ be a group with multiplication map $m: G \times G \rightarrow G$, and let $\pi ,\pi ': G \times G \rightarrow G$ be the projection maps onto the two factors. Then the diagram of sets
consists of pullback squares, but the induced diagram
does not (except in the case where $G$ is trivial).