I am still pretty new to category theory. I am reading the Category for working mathematicians. The author defines the axiom of arrows only meta category as followings.
(1) Composition (k g)f is defined iff k(g f) is defined.
(2) Triple kgf is defined whever both composite kg and gf are defined.
(3) For each arrow g of C, there exist identity arrows u and u' such that u'g and gu are defined.
Isn't (2) is a repetition of (1) as (1) automatically implies (2)? Or is he considering that in some case (kg)f=a and k(gf)=b are defined but (kg) or (gf) is not defined?
What is the relationship between arrows only metacategory with a metacategory from which all objects of arrows only metacategory comes. It seems that the author is trying to imply that these two metacategories define the same metacategory. Correct me if I am wrong. Furthermore, is this some sort definition of functor which looks like functor in arrows' only metacategory's morphism sense.
(1) doesn't imply (2). There's no obvious reason why (1) together with the existence of both kg and gf should imply that (kg)f is defined.
In fact, imagine the algebra consisting of all words of length 1 and 2 in some alphabet, and composition is just concatenation. This algebra satisfies (1), but does not satisfy (2).