$\DeclareMathOperator\id{id}$
$\DeclareMathOperator\Hom{Hom}$
Let $C$ be a strict $n$-category.
Then $C$ has objects, $1$-morphisms, $2$-morphisms,...,$n$-morphisms
I'm considering the concept of an invertible $k$-morphism for $k>1$
We know that $\forall0\leq l<k\leq n$, if two $k$-morphism $f,g$ satisfies that the $l$-target of $f$ coincides $l$-source of $g$, then we can form the composition $g\circ f$ in $l$-direction. In particular, there are $k$ different ways to composite $k$-morphisms. (For example, 2-morphisms have verticle and horizental composition)
It seems that usually we define a $k$-morphism to be invertible if it admits an inverse with respect to composition in $(k-1)$-direction. For example, informally, a gaunt $n$-category is a strict $n$ category that all invertible $k$-morphisms are identity. And the formal definition of gaunt $n$-category is as follows:
Question: In a gaunt $n$-category defined as above, can there be a $k$-morphism which is non-identity, and admits an inverse with respect to composition in $l$-direction for some $0\leq l<k-1$?
I have the following example in mind, but I'm not sure whether this is gaunt in the above sense:
Define a $2$-category as follows:
Object: only one object, called $a$
$1$-morphism: only the identity $\id_a$
$2$-morphism: for each $n\in\mathbb{Z}$, a $2$-morphism $n:\id_a\to\id_a$
Define the composition functor $\Hom(a,a)\times\Hom(a,a)\to\Hom(a,a)$ by:
On object: $(\id_a,\id_a)\mapsto\id_a$
On morphism: $(m,n)\mapsto0$, $\forall m,n\in\mathbb{Z}$
The example in my question isn't a 2-category
And I find that it's not too hard to show that if a $k$-morphism has inverse w.r.t. $l$-composition, then the inverse is also inverse w.r.t. $i$-composition for $l＜i＜k$