I was thinking about presenting categories with nothing but equations over morphisms. I wondered about products. The definition of a product has its genesis in the following diagram shape
You would say that whenever you have this shape, then for any D such that
blah blah...the axioms for the product. I thought about how to do this with just words in the arrows without reference to either source and target or objects. You can start with words like
$dab$ and $dac$
where a:D->A and and b: A->B and c:A->C and d:E->D.
You might say: if, the existence of the equations
implies that $a$ is unique in that if there exists words
then this constitutes a product.
Could something like this work? I mean, can you present a category as just words over morphisms along with equations and a bit of language with quantifiers? Second, could this kind of definition work for products?