EDIT: Are there references to the literature that works out dervied functors for categories enriched over abelian monoids? (I narrowed down the question in the hope for an answer.)

In my book, semigroups have associative multiplication (or let us say addition in the abelian case). Since he assumes a neutral element, I assume he means commutative monoid, in which case I find this question to be a very natural one. If Colin doesn't mean to include associtativity as an axiom, I find the question far less natural.
– Todd Trimble♦Sep 18 '11 at 15:07

(Semigroups -abelian or not- are defined to be associative!)
– QfwfqSep 18 '11 at 15:10

Yeah, I was confused because Colin forgot to list associativity as one of the axioms (unless he really meant the non-associative analogue!).
– Harry GindiSep 18 '11 at 15:20

2

(In case my initial comment seems confusing, it was in response to a comment of Harry which he later deleted.)
– Todd Trimble♦Sep 18 '11 at 20:23

yes. i do mean associative. usually i find the nomenclature "monoid" confusing. i find the analogy between ring and semiring, and group and semigroup.
– Colin TanSep 19 '11 at 2:47

definedto be associative!) – Qfwfq Sep 18 '11 at 15:10