I was reading about formal definition of diagrams in category theory when I came across the definition of finite shape category:
"Assume $A$ is a small category and $B$ is a category. Then a diagram $C$ in $B$ with shape $A$ is the functor $F:A \to B$. The shape $A$ is finite iff there exists a positive natural number $M$ such that for any composition $f_1 \circ ... \circ f_m$ where $m > M$ there exists $f_i$ such that $f_i$ is the identity morphism."
My question is whether the $f_i$'s are allowed to be repeated? Since if they are then this definition does not hold true for many cases as we can simply let all of $f_i$ be a single non-identity morphism.
The morphisms have to be composable. In order to repeat one of the $f_i$, there would have to be a (non-identity) loop in the category, which I think is the sort of thing we're trying to avoid. There can't be any infinite chains of distinct composable morphisms either.
This notion appears to be very similar to the idea of a direct category.