A locally finite category is a category $\mathcal{C}$ such that each arrow can be written as a composition of non-identity arrows in only finitely many ways. To get a picture of what this means, view $\mathcal{C}$ as a graph, where the nodes are objects and the directed edges are all the arrows besides the identity arrows. Note that loops besides identity arrows are permitted. Then a decomposition of an arrow $f\colon A\to B$ is a directed path from $A$ to $B$ in the graph. So the statement that each arrow can be decomposed in only finitely many ways essentially means that there are only finitely many directed paths from $A$ to $B$ in the graph. This analogy only breaks down in the case where $A$ and $B$ are the same object.

The notion of a locally finite category generalizes the notion of a locally finite poset. The condition that each interval $[x,z]$ in a poset $P$ contain only finitely many elements $y$ of $P$ implies that a locally finite poset is locally finite when viewed as a category, since between two objects of $P$ there is at most one arrow.

If $\mathcal{C}$ is a locally finite category, then an algebra over the category arises as the dual of a coalgebra over the category.