TITLE: Unbounded resolutions for Grothendieck abelian categories
QUESTION [12 upvotes]: Consider the following result:

Theorem 1: Let $\mathsf{A}$ be a Grothendieck abelian category. Then every complex in $\mathsf{C}(\mathsf{A})$ has a $K$-injective resolution.

As far as I know, the first proof of this result was in Localization in categories of complexes and unbounded resolutions. In 2003 another proof was published in Resolution of unbounded complexes in Grothendieck categories. There also seems to be a proof in Categories and Sheaves by M. Kashiwara and P. Shapira, but it also relies on some set theory and is quite long. I found no other proofs.
The second proof is possibly the shortest but, due to a lot of set theory involved, seems very strange to me. A more conceptual route seems to be using the Gabriel-Popescu theorem and Brown representability to prove corollary 5.2 in Localization in categories of complexes and unbounded resolutions and then conclude the proof. That's better, but it is still quite long and difficult.
Given the importance of Theorem 1, I wonder if there exist other, perhaps simpler, proofs of this result.

REPLY [11 votes]: It appears that this result may go back to a 1984 letter of Joyal to Grothendieck.  The reference to this letter, as well as some other early references, can be found in Example 3.2 in the paper Cotorsion pairs, model category structures, and representation theory.  Later references using this approach (based on the small object argument) include Corollary 7.1 in the 2007 paper Kaplansky classes and derived categories.  There is also a very recent paper of mine Derived, coderived, and contraderived categories of locally presentable abelian categories, Corollary 8.5.