products in a category without reference to objects or sources and targets Hi,
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
A->B
A->C
You would say that whenever you have this shape, then for any D such that
D->A->B
and 
D->A->C
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 
$dab=e$ and
$dac=f$
implies that $a$ is unique in that if there exists words
$dxb$
$dyc$
then $x=y=a$
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?
 A: As Freyd and Scedrov show in "Categories, Allegories", you can think of a category as some kind of partial monoid. Such a monoid has a set of partial units that take over the role of objects in standard presentations of categories.
A: Yes this is possible. But you have to restrict to "composable" words.
A category may be viewn as a monoid whose composition law is only defined partially: We have a set of morphisms $M$ and maps $s,t : M \to O$ (source/target), then $mn$ is defined as soon as $t(n)=s(m)$ and satisfies $t(mn)=t(m), s(mn)=s(n)$. But this "algebraic point of view" is often not so enlightening. It is better to picture your category as sort of a graph with a notion of commutative diagram. The product of two objects $p,q$ is the "smallest" object $p \times q$ which lies "above" $p,q$ in the sense the there are two arrows $p \times q \to p$ and $p \times q \to q$. You should really draw this.
Just to compare, here is the algebraic definition: A product $p \times q$ is a pair of $m,n \in M$ with $s(m)=s(n), t(m)=p, t(n)=q$, such that for each other such pair $(m',n')$, there is a unique $u \in M$ such that $s(u)=s(m'), t(u)=s(m)$ and $m'=mu, n'=nu$.
