In the mathematics of graph theory, two graphs, G and H, are called homomorphically equivalent if there exists a graph homomorphism
Homomorphic equivalence also comes up in the theory of databases. Given a database schema, two instances I and J on it are called homomorphically equivalent if there exists an instance homomorphism
In fact for any category C, one can define homomorphic equivalence. It is used in the theory of accessible categories, where "weak universality" is the best one can hope for in terms of injectivity classes; see