As we know by definition, a functor for example $T\colon R\textrm{-}\mathsf{Mod}\to\mathsf{Ab}$ (from the category of $R$-modules to the category of abelian groups) is "exact" precisely when for any short exact sequence: $$0 \to A \to B\to C\to 0$$ the sequence $$0\to T(A) \to T(B) \to T(C) \to 0$$ is also exact.
My question is: Is this equivalent to the condition where these sequences do not begin with zero? In other words can we say $T\colon R\textrm{-}\mathsf{Mod}\to\mathsf{Ab}$ is exact precisely when for any exact sequence: $$A \to B \to C,$$ the sequence $$T(A) \to T(B) \to T(C)$$ is also exact?
Obviously, the latter implies the exactness definition, but I am not yet convinced about the other direction.