A category $C$ is $L$-finite if the following equivalent conditions hold:
The notion of L-finite category is a sort of categorification of the notion of K-finite set:
A set $X$ is $K$-finite if the top element $1 \in \Omega^X$ belongs to the closure of the singletons under finite unions.
A category $C$ is $L$-finite if the terminal object $1\in Set^C$ belongs to the closure of the representables under finite colimits.