I'm looking for, or hoping to inspire the creation of, a list of conventional names for categories that come up often.

For example, we have *the terminal category* $\fbox{$\bullet$}$, a nice name. I've heard this category $\fbox{$\bullet\to\bullet$}$ called *the free arrow category*. That's fine by me. What about the category with one object $x$ and an one arrow $p^n\colon x\to x$ for each natural number $n$ (and the obvious composition law)? It's hard to draw it without a package; here's my best attempt: $\fbox{$\bullet\circlearrowleft$}$. I'd like to call this *the free loop category*. But is that standard?

The categories $[n]$ for $n\in\mathbb{N}$ might be drawn $\fbox{$\bullet^0\to\bullet^1\to\cdots\bullet^n$}$. I might call this *the length-$n$ chain category.* What about $\fbox{$\bullet\rightrightarrows\bullet$}$? I might call this *the parallel arrows category*. Would one know what I meant by *the two equalized arrows category* or *the two coequalized arrows category*? Hint: they each have three objects and four non-identity morphisms. But what if I didn't want a certain commutative diagram to hold there, i.e. I wanted to name the related five-morphism categories?

The point I hope is clear. Does anyone know of a definitive list of names for important diagram categories?

Thanks!