Primacy of arcs/arrows over vertices/objects

Qu 1: At about the time of Freyd's book there were two approaches to defining categories. One came from algebraic topology and homological algebra, thus from Eilenberg and MacLane and used the objects and arrows definition, the other was motivated by differential geometry and used the arrows only formalism. This second one is perhaps better suited to those areas where the arrows are what is seen first. For instance, when introducing the concept of the fundamental groupoid of a space then you can think of the idea as a set with a partially defined composition satisfying certain rules and that has distinct advantages for that setting. Ehresmann wrote his book on categories from this viewpoint, but some of the basic ideas do end up being less clear from that viewpoint, others are perhaps clearer. That approach is also linked to more algebraic ideas such as inverse semigroups.

The first 'objects plus arrows' approach is thought to be more accessible to researchers with a `standard' mathematical background e.g. from algebra or algebraic topology. There the objects are usually thought of as being what is being studied and the morphisms are a tool for that study.

The recent use of categorification, quasicategories, internal categories, enriched categories and other similar ideas, tends to show that both viewpoints are best kept in balance, meeting in that higher categorical area.

Put simply I thing the answer to your question is: historically the comparison of objects using arrows was the more pressing application to start with. It came to dominate. More researchers came to use it.

Now the question arises `which to use with a given audience? and that is a hard one to answer.

The first obvious answer is that the intuition behind categories involves objects, so there is no need to remove them from the definition, even if the resulting formulation turns out to be more economic. Try teaching a second or third-year student categories without mentioning objects...

A second possible reason is that it may be involved to ask that morphism between two objects form a set, but objects are allowed to be a proper class. Without mentioning objects, one should say that arrows can form a proper class, but every way I can think of to ask for Hom(A, B) to be a set really boils down to putting objects back in the picture.