Cancellation of direct products of digraphs

We investigate expressions of form A×C ≅ B×C involving direct products of digraphs. Lovász gave exact conditions on C for which it necessarily follows that A ≅ B. We are here concerned with a different aspect of cancellation. We describe exact conditions on A for which it necessarily follows that A ≅ B. In the process, we do the following: Given an arbitrary digraph A and a digraph C that admits a homomorphism onto an arc, we classify all digraphs B for which A×C ≅ B×C