The first quotes is just because one can talk about K theory in various levels of generality and this one is something of a midpoint: more general than just for ordinary rings, but not all the way to handling arbitrary Wauldhausen ∞-categories that one might want for stating devissage.
The "a" was just sloppy writing, K theory at this level of generality is an additive and group-like functor... the unique such functor receiving a map from core!
(I definitely don't read this as nitpicking, it's good to clarify what's going on precisely :) )