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).

dropcomposition then you have what Mac Lane calls a precategory (and everyone else, a multidigraph). If you have compositionandidentities then you have what Lambek and Scott calls a deductive system. $\endgroup$