Proposition 3.5 of "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts claims that the is an equivalence of categories $$ D^b(coh(X))\overset{\sim}{\to} D^b_{coh}(Qcoh(X)) $$ where $D^b(coh(X))$ is the derived category of bounded complexes of coherent sheaves, and $D^b_{coh}(Qcoh(X))$ is the derived category of bounded complexes of quasi-coherent sheaves with coherent cohomologies.

However, the proof of Proposition 3.5 only shows that the functor is essentially surjective. We know that a priori there are more morphisms in $D^b_{coh}(Qcoh(X))$ than in $D^b(coh(X))$. Then how to show that the morphism sets are the same?