A homological algebra theorem states
Theorem: Let $T: \mathscr{A} \to \mathscr{B}$ be a left exact functor between abelian categories, and let $X^\bullet \xrightarrow{f} Y^\bullet$ be a quasi-isomorphism of chain complexes of $T$-acyclic objects in $\mathscr{A}$. Then $T(X^\bullet) \xrightarrow{T(f)} T(Y^\bullet)$ is a quasi-isomorphism.
In the proof in the book I'm reading (Iversen's Cohomology of Sheaves), the first line is
Consider the mapping cone to see that it suffices to treat the case where $Y^\bullet =0$.
I don't get it...can anyone give me a clue here? I'm just looking for help on this first statement.
Any additive functor between abelian categories preserves mapping cones, and a map is a quasi-isomorphism iff its mapping cone is acyclic, or equivalently quasi-isomorphic to $0$.