Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear naturally as a class of morphisms with a partial composition rule, without an explicit or evident class of objects? Are there examples which really convince us that the morphisms-only-definition is natural? This may be a vague question, but in principle I would like to know if there is any pedagogical value in the morphisms-only-definition of a category (apart from its usefulness for e.g. mathematical logic).
I think that there many groupoids which appear only via their morphisms. In fact, the original conception of a groupoid was that of a 'group' with a partially defined group multiplication (curiously, it seems that categories are usually not seen as 'monoids' with a partially defined multiplication). Therefore I would like to favor examples which are no groupoids.