Of course, using identity morphisms and composition, we can turn one into the other; which is more fundamental depends on which shapes you prefer.
Eric: Are there any consistency requirements for a 2-morphism? For example, in the bigon above, if , , and , are there requirements on regarding and ? For example, should come with component 1-morphisms and such that
? Could there be a 2-morphism without the corresponding 1-morphism components?
Urs Schreiber: in any given 2-category you have to specify which 2-morphisms exactly there are supposed to be, what exactly you allow between and . When you ask about components, it seems you are thinking of 2-morphisms specifically in the 2-category Cat. Here, yes, the allowed 2-morphisms are those that are natural transformations between their source and target 1-morphisms, which are functors.