I have a question regarding composition of exact functors in $Abelian$ categories. lets say I have the two right exact functors: $F: \mathscr C_1 \to \mathscr C_2$, $G: \mathscr C_2 \to \mathscr C_3$.
Im wandering whether the composition of the two functors $G \circ F$ is also right exact?
I'm asking because it sounds right by intuition, but when I go by the definitions its not really clear: If I understand correctly, a right exact functor transfrom a SES $0 \to A \to B \to C \to 0$ to an exact sequence $F(A) \to F(B) \to F(C) \to 0$.
But now, i left with an exact sequnce, ant not a SES, so how do I know that the functor G transform that exact sequence into the exact sequence $G(F(A)) \to G(F(B)) \to G(F(C)) \to 0$?