Recent algorithmic advances in algebraic automata theory drew attention to semigroupoids (semicategories). These are mathematical descriptions of typed computational processes, but they have not been studied systematically in the context of automata. Here, we use relational programming to explore finite semigroupoids to improve our mathematical intuition about these models of computation. We implement declarative solutions for enumerating abstract semigroupoids (partial composition tables), finding homomorphisms, and constructing (minimal) transformation representations. We show that associativity and consistent typing are different properties, distinguish between strict and more permissive homomorphisms, and systematically enumerate arrow-type semigroupoids (reified type structures).