What's the name for a function which translates faithfully into a quotient, i.e. commutes with the quotient?
I'll use the example of a group. Let $X$ be a group and $X/Y$ be its quotient.
Let $f:X\to X$ have the property that $\forall x\in X: \forall \{xy:y\in Y\} :f(xy)=f(x)y$
As an example, in $\Bbb Z/n\Bbb Z$ the function $f(x)=x+n$ has this property.
I always say that $f$ commutes with the quotient. Is there a more widely used terminology for this? And is there a terminology e.g. outside of groups - in any quotient, e.g. in a monoid or a topological quotient space?
This always implies that $f:X/Y\to X/Y$ is well-defined as a function.