Equivalent definition of exactness of functor?

Solution 1:

Any exact sequence can be broken down into short exact sequences (the $C_i$ are kernels/images):

So, since your functor $F$ preserves short exact sequences, you can apply $F$ and the diagonal sequences will remain exact. It's now a general fact that in any such diagram, if the diagonals are exact, then the middle terms are exact as well (by diagram chasing).

EDIT: If $f_i\colon A_i\to A_{i+1}$, then $C_i=\ker(f_i)$ which by exactness is isomorphic to $\operatorname{im}(f_{i-1})$.

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