This is a rather vague question, but perhaps we can talk about it.
There are two types of mathematical objects (which don't exclude each other):
A) There is a good description of morphisms defined on this object.
B) There is a good description of morphisms defined into this object.
Thus A) means that the covariant hom-functor is understood, and B) means that the contravariant hom-functor is understood. This applies most notably to universal objects. Within category theory, the concepts are just dual to each other and so the "theory" of A) is essentially the same as the theory of B). But most categories studied in mathematics don't come together with their dual, so that this categorical argument is not really good. In fact, I have the feeling that in 'daily mathematics', A) appears much more often than B). And that it is easier to work with them. Of course, we could argue about that. For example, I have a better feeling with colimits than with limits. [perhaps I will add examples here]
If you have the same feeling: Can we give reasons for this?
I think that the basic principle of gluing, which appears in many geometric categories, always belongs to A). This could be a reason. What do you think?