Term for this concept in category theory? Suppose we have three objects $X,Y,Z$, and a morphism $m:X\to Y$.
Moreover, this morphism has the following property:
For any morphism $f:Z\to Y$, there exists a morphism $f_X:Z\to X$, such that $m\circ f_X=f$.
Intuitively, this seems to me to capture the notion that “any information we need to pick an element of $X$, there is enough information in Y to do so”
Essentially, it seems to me that this generalizes the idea of a surjective function, but this concept is already generalized by “epimorphism”, whose definition is different.
Is my definition equivalent to that of epimorphism? If not, is there a term for my definition?

The natural way to name this property is "the object $Z$ has the left lifting property with respect to the morphism $m$". Indeed, if the category has an initial object, then the property you mentioned is equivalent to the left lifting property between $i_Z$ and $m$, where $i_Z$ is the unique morphism from the initial object to $Z$. If, in addition, the lifting morphism $f_X$ is unique for every $f$, then this property is called "the object $Z$ is orthogonal to $m$" (denoted by $Z\perp m$, see definition 5.4.2 in F.Borceux, "Handbook of Categorical Algebra 1").
