A cyclic group $G$ is a group such that there is an element $f$, such that for every $g\in G$, we have that $g=f^z$ for some $z\in \mathbb Z$.
Let $C$ be a category with objects $X,Y$. We can call the homset $\text{Hom}(X,Y)$ a “cyclic homset” if there is an element $f:X\to X$ and $h:X\to Y$, such that for every element $g\in \text{Hom}(X,Y)$, we have that $g=h\circ f^n$ for some $n\in \mathbb N$.
Does this concept exist? Do cyclic homsets have similar properties as cyclic groups or cyclic monoids?