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?