Injective cogenerator of abelian groups ($\mathbb{Q}/\mathbb{Z}$)

The divisible group $D=\mathbb{Q}/\mathbb{Z}$ is injective: every homomorphism $n\mathbb{Z}\to D$ extends to $\mathbb{Z}$ by choosing an $n$th division of its value at $n$, and the Baer criterion applies. It is a cogenerator because any nonzero element generates a cyclic subgroup admitting a character nonzero on that element, and injectivity extends that character to the whole group. Consequently a map $u:T\to U$ of abelian groups is injective exactly when restriction $U^{\ast }\to T^{\ast }$ is surjective: injectivity gives extension of characters, while a nonzero kernel element is detected by a character that cannot extend along $u$.