By a total (or left-total) relation I mean a binary relation $R \subseteq X \times Y$ where there is, for each $x \in X$, at least one $y \in Y$ with $(x,y) \in R$. Equivalently stated, I mean relations under which the projection to the first coordinate is a surjection.
I would now like to look at the category where
- the objects are all sets,
- the morphisms between two sets $X,Y$ are all total relations $R \subseteq X \times Y$ and
- the composition of two total relations $R \subseteq X \times Y$ and $S \subseteq Y \times Z$ is given as follows: $S \circ R := \{(x,z) \mid \exists y \in Y: (x,y) \in R, (y,z) \in S\}$.
What is the product of two sets in this category (if it exists at all)? I know that it would be the disjoint union if one took ALL relations as morphisms, but I think that this is not the case here.
Can anyone help?