Let $0\to A\to B\to C\to 0$ be a short exact sequence of finite abelian groups and $Hom(-,A)$ functor induces an exact sequence $0\to Hom(C,A)\to Hom(B,A)\to Hom(A,A)$. How to prove the equivalence bewteen the split exactness of the original sequence and the right exactness of the induced sequence?
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Please be more specified on your question: are you asking that the split exactness of $0 \to A \to B \to C \to 0$ is equivalence to the exactness of $0 \to Hom(C,A) \to Hom(B,A \to Hom(A,A) \to 0$? – SalutaFungo Oct 17 '23 at 02:49