Exactly the same as in the objects-and-arrows presentation of category!
Working with the “arrows-only” definition of a category doesn't mean you can't talk about objects, it just means that they're themselves a defined notion. Some constructions can be very nicely given in purely arrows-only language, but for many things — and I think product is one — it seems most natural to define “objects” and then to use them in the statements of further definitions.
You can certainly then unfold the definition to give it in a way that doesn't mention objects. But I think it's important to note that you don't need to do this, and a priori, no obvious big reasons one would want to!
Edit: Reading Martin B's comment, I realise I may well have misunderstood the intent of your question. I'm leaving this answer, though, as I think it's still a point worth making!