The standard way to encode a group as a category is as a "category with one object and all arrows invertible". All of the arrows are group elements, and composition of arrows is the group operation.
A loop obeys similar axioms to a group, but does not impose associativity. Inverses need not exist, but a "cancellation property" exists -- given $xy = z$, and any two of $x$, $y$, and $z$, the third is uniquely determined.
Quasigroups need not even have a neutral element.
Given the lack of associativity, arrows under composition do not work to encode loop elements.
Is there a natural way to do this?