Let $X$ be some smooth algebraic variety. I would like to understand the relation between the following two categories:

1.  $D^b_{cd,1}\text{Coh}(X) \subset D^b\text{Coh}(X)$: the full subcategory of the derived category of coherent sheaves with cohomologies being supported in codimension $\geq 1$.

2. $D^b\text{Coh}_{cd,1}(X)$: The derived category of the abelian category of coherent sheaves supported in codimension $\geq 1$.  

I was trying to show that for a bounded complex $F^\bullet$ of coherent sheaves with cohomologies supported in codimension $\geq 1$ there exists a quasi-isomorphic complex $E^\bullet$ with $E^i$ being supported in codimension $\geq 1$, which would show that they are equivalent, but I have ran into some problems like $i_*\circ i^*$ for $i$ a closed immersion not being exact (the idea was to take the union of the support of cohomologies and restrict the complex to this union). Is it still true that these categories are equivalent? If not, are they at least related in some nice way?