TITLE: Is the bounded derived category of coherent sheaves of a variety a small category?
QUESTION [7 upvotes]: The question is in the title. 
I am trying to apply the Mitchell (Freyd-Mitchell?) embedding theorem, which states that for every small abelian category $A$, there exists a ring $R$ such that A embeds into the category $R$-mod. The derived category is not abelian, of course, but I have a particular subcategory that is abelian, and life would be easiest if the derived category was smal, so that the subcategory was small and abelian.

REPLY [5 votes]: The answer to your question really depends on what you mean by the word "the".  An unhelpful answer is that the coherent sheaves over any variety form a proper class (hence "no").  A more useful answer is (as mentioned in the comments) that there exists a small category that is equivalent to any category that can be reasonably called the bounded derived category of coherent sheaves (hence "yes").
Furthermore, the construction of such a category can be accomplished without the use of replacement.  In particular, the category lives in the same ZC universe (i.e., $V_\alpha$ for $\alpha$ a not-necessarily-inaccessible limit ordinal greater than $\omega$ - see e.g., Wikipedia or the set theory section of the Stacks project) as the defining field.