MartinEscardo, 
It is just me? The following definition of category hurts my categorical instincts, because it uses object equality.
A category consists of
A collection of objects.
A collection of morphisms.
Each morphism f has two assigned objects, its source s(f) and its target t(f).
For each pair of morphisms f,g such that t(f)=s(g) there exists a specified morphism g ∘ f such that [it doesn't matter what]
[Some axioms are satisfied.]
It is (4) that hurts my categorical instincts.
There is no reason to have "evilness" (in the categorical sense, rather than the emotional sense) built-in in the definition of category!
This definition is, for example, adopted by Freyd.