I'm writing "a structure preserving surjection" way too much when I need to refer a function of the following property:
$ Y \subseteq Z, X \subseteq Z. g: Z \to A, g \text{ is some fixed function}.$ $ \phi : X \to Y, \phi \text{ is a surjection, and } g(\phi(x)) = g(x).$
Would a category theorist frown if I call $\phi$ a epimorphism?