Lemma 3.53 Rotman's "An Introduction to Homological Algebra"

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In Rotman's book An Introduction to Homological Algebra, in the proof of Lemma 3.53, enter image description here
I think we don't need the fact that $\mathbb Q/\mathbb Z$ is injective, because we don't have $0$'s in the extremities of the sequence $A\rightarrow B\rightarrow C$.