In the derived category $D^b(\mathcal{A})$ of an abelian category $\mathcal{A}$, obtained by taking the Verdier quotient wrt. all quasi-isomorphism, a morphism is given by a roof (or span) $ X\stackrel{\sim}{\leftarrow} A \to Y$. The composition with a morphism $Y \stackrel{\sim}{\leftarrow} B \to Z$ can be written as the roof
$$\begin{matrix} &&&& A\times_Y B\\ &&& \swarrow && \searrow\\ && A &&&& B\\ & \swarrow && \searrow && \swarrow && \searrow\\ X &&&& Y &&&& Z.\end{matrix}$$
This derived category $D^b(\mathcal{A})$ can be seen as the homotopy category of the model category $\mathrm{Ch}(\mathcal{A})$, which has quasi-isomorphisms as weak equivalences. In general, however, homotopy categories need not have roof as morphisms, but general zig zags $X \stackrel{\sim}{\leftarrow} A_1\to B_1 \stackrel{\sim}{\leftarrow} A_2\to B_2 \stackrel{\sim}{\leftarrow} \dotsb \to Y$.
How do I know when it is enough to consider roofs only? I would have guessed that it is enough if the model category has pullbacks and those preserve weak equivalences.
But I have been told that the derived category dga algebras does not have only roots as morphisms—this contradicts my expectations, as dga algebras should have pullbacks preserving quasi-isomorphisms—or don't they?