Let $C$ be a category. Then $C$ is said to be direct if $C$ has no infinitely long descending sequence $$ \cdots\to\bullet\to\bullet\to\bullet $$ of non-identity morphisms in $C$.
Let $\mathrm{ON}$ denote the class of all ordinal numbers (in the sense of von Neumann), equipped with the canonical irreflexive order $\in$, which makes $\mathrm{ON}$ a category. The category $C$ is direct if and only if it admits a identity-reflecting functor $C\to\mathord{\mathrm{ON}}$, or more explicitly, a map $\phi\colon\operatorname{Ob}(C)\to\mathord{\mathrm{ON}}$ such that any non-identity morphism $a\to b$ satisfies $\phi(a)<\phi(b)$. Functions of this kind seems (but not surely) to be called "degree functions", and makes $C$ a Reedy category with trivial degree-lowering component. The names for directness and degree functions are taken from nlab page for "direct category".
Now let $C$ be a direct category. Then there is the "smallest" degree function $\mathrm{ht}\colon C\to \mathrm{ON}$, in that $$ \forall \phi\colon\text{degree function on $C$},\, \forall a \in\operatorname{Ob}(C),\, \operatorname{ht}(a) \leq \phi(a) \text{.} $$ This is easily defined by the well-founded induction on $\operatorname{Ob}(C)$ by $$ \operatorname{ht}(a) := \left\{\operatorname{ht}(b)\,\middle|\,\text{There is a non-identity morphism }b\to a\right\}\text{.} $$ Question: Is there a name for this function? This has the name height function (for posets) in order theory, but I could not find in the literature the terminology for the "height function" in this sense. I need to use this function in my paper, and I am not sure I am allowed to use the word "height" for this kind of concept.
You can have a look at this article by Mike Schulman, especially definition 4.6 where height is also used (for inverse categories but you can dualize that).