This seems challenging because of the apparent non-transitivity of the covering relation (defined below) which would violate the category axioms.
@Berci suggests here that one potential generalization is to call arrow covering whenever =ℎ implies or ℎ is an isomorphism. But I would need more explication to understand this possibility (and to extend it to a property of the entire category).
If this question has an answer, I believe that it offers another way to express how "veridical" a functor is, beyond its fullness of faithfulness, as suggested here
Defn: The covering relation (from here).
Let $X$ be a set with a partial order $\le$. As usual, let $<$ be the relation on $X$ such that $x<y$ if and only if $x\le y$ and $x\neq y$.
Let $x$ and $y$ be elements of $X$.
Then $y$ '''covers''' $x$, written $x\lessdot y$, if $x<y$ and there is no element $z$ such that $x<z<y$. Equivalently, $y$ covers $x$ if the interval $[x,y]$ is the two-element set $\{x,y\}$.
When $x\lessdot y$, it is said that $y$ is a cover of $x$. Some authors also use the term cover to denote any such pair $(x,y)$ in the covering relation.