Simple objects in higher categories

Question

$\newcommand{\End}{\mathrm{End}} \newcommand{\id}{\mathrm{id}}$ In an $n$-category ($n=1,2$) Reutter defines a simple object in the following way:

An object $X$ is simple, if one of the following equivalent conditions hold:

1. every subobject $A\hookrightarrow X$ is either zero or an equivalence
2. $X$ is indecomposable under $\oplus$ (working in a semisimple category)
3. $\End_{\mathcal{C}}(X)=\mathbb{K}$ when $n=1$, while $\End_{\mathcal{C}}\left(\id_X\right)=\mathbb{K}$ when $n=2$. (for an algebraically closed field $\mathbb{K}$).

I would like to understand how this definition generalises to arbitrary $n$.

1. Condition 1 is the traditional definition of a simple object in a category so that is one way to define simple objects for arbitrary $n$. However, it does not buy me much understanding of what a simple object actually is.
2. It makes sense that condition 2 also directly generalises to arbitrary $n$. This is the essence of an object being called "simple".
3. My main question concerns condition 3. By the looks of it I would expect that in an $n$-category an object $X$ is simple if $$ \End_{\mathcal{C}}\underset{(n-1)\ \id\text{'s}}{\underbrace{\left(\id_{\id_{\ddots_{\id_X}}}\right)}}=\mathbb{K}. $$ Is that definition correct? In what way is it equivalent to the other two conditions?