Is there a standard category-theoretic way to express a loop or quasigroup? 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?
As expressed by Qiaochi Yuan in this and this comment, the way that category theory applies to studying loops and quasigroups is in the form of a category whose objects are loops, resp. quasigroups.

Only a few structures can actually be described as categories having certain special properties (among which sets, groups, partially ordered sets). For a structure to have any chance of being a "special type of category", it is of course necessary that the defining properties for a category are somehow satisfied by the structure in question.

For loops and quasigroups, this is not obvious to say the least, due to the lack of associativity (which is all-important in category theory).
I know this is an older thread but I've been thinking about basically this question for a few months now
Something I've come up with is that a quasi-group requires certain automorphisms to exist on the categorical product of 3 elements.
An example of this, in the Category of sets the products AxB, CxA and BxC are all isomorphic to subsets of AxBxC which is isomorphic to CxAxB and BxCxA.
These isomorphisms are what allow us to distinguish between a binairy function and it's left and right inverses, and are generalizable withing the context of categorical products (I believe) as the proper morphisms should exist between categorical products of pairs and products of triples.
Once these automorphisms are established, I believe quasi-groups arise naturally from the existence of morphisms between categorical products of pairs and objects.
This is all really sketchy right now, I'm honestly just getting into category theory but I believe the above works?