Does anyone know if there exists a name in the literature for the data of
1) a class of objects,
2) for each pair of objects $(x, y)$ a set $hom(x, y)$
3) for each triple of objects $(x, y, z)$ a morphism of sets $hom(x, y) \times hom(y, z) \to hom(x, z)$.
I don't impose any conditions on this data (if I were to impose the usual associativity and identity axioms this would be the definition of a category).
Best Answer
As Zhen Lin already noted in his comment, this is called a deductive system in section I.1 of
J. Lambek, P. J. Scott, Introduction to higher order categorical logic
I think that the idea is the following: We treat a morphism $A \to B$ as a deduction from $A$ to $B$. The identity morphism is the trivial deduction $A \to A$ and the composition $A \to B$, $B \to C$ $\leadsto$ $A \to C$ is a rule of inference, namely the hypothetical syllogism.