I recently faced the concept of derived category, introduced as localization of homotopy category. I tried to verify of the axioms and I stucked in the following: In the definition of multiplicative system I see that LMS3 of https://stacks.math.columbia.edu/tag/04VB (Stacks Project) is required. This axiom is widely used in proving that the composition of two maps is well defined.
However, in my notes, I'm unable to prove (there's no proof of the fact there), just the following remark:
In the literature one usually finds an additional axiom for multiplicative systems called the cancellation property. It was not used in the above construction; it is necessary when one wants to prove the stronger property that the Hom-sets in the category $S^{−1}C$ arise as filtered direct limits of Hom-sets in $C$.
Beside not understanding this sentence, is it true that right cancellation can be omitted or required? If so, how one can prove that the composition is well defined?
Any help or clarification would be appreciated.