# A question on the proof of $D^b(coh(X))\simeq D^b_{coh}(Qcoh(X))$

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?

Indeed, the proof in the book shows that if $$G$$ is a bounded complex of quasi-coherent sheafs with coherent cohomologies, then there is a subcomplex $$G_1\subseteq G$$ with same cohomologies of $$G$$, where $$G_1$$ is bounded complex of coherent sheafs. Then you can prove directly that two bounded categories have same hom set.
In fact, here $$K^b(Coh(X))$$ is cofinal subcategoy of $$K^b(Qcoh X)$$. In general, the following is true:
Let $$S$$ is a multiplicatively closed subset of an additive category $$\mathcal A$$, suppose $$\mathcal B$$ is full subcategory of $$\mathcal A$$, if $$S\cap \mathcal B$$ is also a multiplicatively closed subset of $$\mathcal B$$. If for any $$s: X\rightarrow Y$$ in $$S$$, where $$X\in \mathcal A$$ and $$Y\in \mathcal B$$, there exists $$t:Z\rightarrow X$$ with $$Z\in \mathcal B$$ such that $$st\in S$$, then $$(S\cap \mathcal B)^{-1}\mathcal B\rightarrow S^{-1}\mathcal A$$ is fully faithful.