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How can I prove that if

$$0\longrightarrow\mathrm{Hom}(M,A)\xrightarrow{\;\;i_*\;\;}\mathrm{Hom}(M,B)\xrightarrow{\;\;j_*\;\;}\mathrm{Hom}(M,C)$$ is left exact, then $$0\longrightarrow A\xrightarrow{\;\;i\;\;} B\xrightarrow{\;\;j\;\;} C$$ is left exact. I have seen proofs showing that if the second chain is left exact, then the first chain is left exact, but what about proving the converse without depending on the projective module concept.

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    Your notation is confusing, can your please state a little more clearly what you are asking? What are $i,j$? Are they functors?2012-12-28
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    Use the Yoneda lemma.2012-12-28
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    Closely related http://math.stackexchange.com/questions/235372/homomorphism-exact-sequences2012-12-28

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