Let $X$ be a scheme. Let $E$ be a perfect complex of coherent sheaves on $X$ and suppose it admits two global resolutions $ F$ and $F'$. By global resolution I mean that both $F$ and $F'$ are quasi-isomorphic to $E$ and both are complexes of vector bundles.

$\bf{Question:}$ Is it possible to find a global resolution $H$ of $E$ such that $ F, F'$ map mono-morphically into $H$?

I know how to do this when $E$ has perfect amplitude $[0, 1]$ but I don't see how to generalize to say $[0, n]$ for arbitrary $n$.