# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Warning 3.4.4.2. The conclusion of Exercise 3.4.4.1 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

$\xymatrix@R =50pt@C=50pt{ G \ar [d] & G \times G \ar [l]_-{\pi } \ar [r]^-{\pi '} \ar [d]^{m} & G \ar [d] \\ \ast & G \ar [l] \ar [r] & \ast }$

consists of pullback squares, but the induced diagram

$\xymatrix@R =50pt@C=50pt{ G \ar [d] \ar [r] & G \coprod _{ G \times G} G \ar [d] & G \ar [d] \ar [l] \\ \ast \ar [r] & \ast \coprod _{G} \ast & \ast \ar [l] }$

does not (except in the case where $G$ is trivial).