= Injective cogenerator of abelian groups
{title2=$\mathbb Q/\mathbb Z$}
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 $U^*\to T^*$ is surjective: injectivity gives extension of characters, while a nonzero kernel element is detected by a character that cannot extend along $u$.
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