Are two homotopic maps between chain complexes equal up to composition with a homotopy equivalence?

Question

Let $f,g : A\rightrightarrows B$ be two maps between chain complexes $A,B$ with terms in an abelian category. Suppose $f,g$ are homotopic. Must there exist a homotopy equivalence $\alpha : C\rightarrow A$ such that $f\circ \alpha = g\circ \alpha$ on the nose? (also, the same question with $\beta : B\rightarrow C$ such that $\beta\circ f = \beta\circ g$)