I've seen that there was a [single-sorted definition](https://ncatlab.org/nlab/show/single-sorted+definition+of+a+category) of a category. In some ways, it seems more understandable than the original definition.


I don't know much about category theory. But I would like to know how each definition is useful.



Why is an object not defined as identity morphism?
They're so automatically interchangeable that it doesn't really make sense to say that one is more useful than the other except in minor ways: if we really care about "symbolic parsimoy" then the object-free approach has a minor advantage, while if we care about matching informal discourse then objects are generally essential (per Andreas' comment).


Ultimately I'd say that the second point wins out - it's hard to beat intuitive clarity if the cost is so minor. But since the translations in both directions are really so trivial, you can use whichever one you want. And to be fair, it's not the case that the two-sorted approach is *always* more intuitive - when thinking of a group as a category, having to consider an utterly pointless object is a bit weird at first. *(EDIT: Simon Henry's comment points out a more convincing example of this, where the two-sorted approach results in meaningfully annoying technical overhead.)*


