proving isomorphism

How to show a chain map is a quasi-isomorphism?

easy, just provide the map in the reverse direction. So, you have a functor and you want to show that it is fully faithful. You got two objects, $X,Y$, send them over to $FX, FY$, there is an injective map $$ \Hom(X,Y) \to \Hom(FX, FY) $$ and we know the image. We probably also know the inverse chain map, just map the top of the chain complex back. small to big back to small is OK, big to small then to big, is hard to say why it is isom. We better have a homotopy explaining the difference.