I wonder whether in any Abelian category $\mathcal{C}$ when we have a long exact sequence $0\to M_1\to M_2\cdots\to M_n\to 0$ and a (covariant) left exact functor $F$ we have $0\to FM_1\to FM_2\to \cdots FM_n$.
I know this is certainly true if $n=3$ (i.e. we have a short exact sequence), but what about in general, may I ask?
This question comes from Page 118 of Assem, Simson and Skowronski's book, Elements of the representation theory of associative algebras, Volume 1, where the authors seem to have used this supposed property in the proof of the Auslander-Reiten Formula.