The Stacks project

Maybe this is supposed to be clear, but regarding the comments just before this definition:

It seems to me that some additional argument is being made to compare two K-flat resolutions, $\mathcal{K}^{\bullet}_1 \rightarrow \mathcal{F}^{\bullet}$ and $\mathcal{K}^{\bullet}_2 \rightarrow \mathcal{F}^{\bullet}$, in addition to Lemma 21.17.12. I guess one could argue by dominating both by K-flat resolutions by another, i.e. $\mathcal{K}^{\prime \bullet} \rightarrow \mathcal{K}^{\bullet}_1$ and $\mathcal{K}^{\prime \bullet} \rightarrow \mathcal{K}^{\bullet}_2$, which are compatible up to homotopy with the maps to $\mathcal{F}^{\bullet}$. Or maybe you have something else in mind?

By considering a K-flat resolution of the other factor $\mathcal{G}^{\bullet}$, I claim that we can see that the resulting isomorphism of functors does not depend on the choice of $\mathcal{K}^{\prime \bullet}$.

That is, the previous sentence could read:

By Lemma 21.17.12, the resulting functor (up to canonical isomorphism) does not depend on the choice of the K-flat resolution.

This canonical isomorphism is compatible with composition, from $\mathcal{K}^{\bullet}_1$ to $\mathcal{K}^{\bullet}_2$ to $\mathcal{K}^{\bullet}_3$, if that makes sense.