definition of left (right) Exact Functors Let $P,Q$ be abelian categories and $F:P\to Q$ be an additive functor. Wikipedia states two definitions on left exact functors (right dually):


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*$F$ is left exact if $0\to A\to B\to C\to 0$ is exact implies $0\to F(A)\to F(B)\to F(C)$ is exact.

*$F$ is left exact if $0\to A\to B\to C$ is exact implies $0\to F(A)\to F(B)\to F(C)$ is exact.


Moreover, it states that these two are equivalent definitions. I'm quite new at this topic so I'm not sure if this is immediately clear or not. Surely, 2. $\implies$ 1., being the more general case. But I don't see how to even approach the other direction; is this merely tautological?

Hint: Let $0\to A\overset f\to B\overset g\to C$ be exact. It means that $f=\ker g$, now consider the canonical factorisation of $g$, letting $C':={\rm im\,} g$, and $g':B\to C'$ (in a category of modules it is just $g'(b):=g(b)$ for all $b$). Then, we have that $0\to A\overset f\to B\overset {g'}\to C'\to 0$ is exact. Apply the hypothesis on this.
