Categories with restricted identities

In this draft (to be moved into the online book in the future, but the draft is nearing finishing this topic, not including functors between categories with restricted identities) I described axioms and properties of categories with restricted identities.

Basically, a category with restricted identities is a category $\mathcal{C}$ together with morphisms $\mathrm{id}^{\mathcal{C}(A,B)}_X$ which are strictly less (in our order of morphisms) than identities $1^A$. These “restricted identities” conform to certain axioms.

Using restricted identities, it is possible to turn a category into a semigroup, which I call “semigroup of unfixed morphisms”, because semigroups elements don’t have “fixed” source and destination objects, but describe common properties of morphisms with different sources and destinations (abstracting objects of the category away).

I wrote all this with the purpose to define “unfixed funcoids” and “unfixed reloids”, to allow abstract away the source and destination of say a funcoid, making it similar to “arbitrary binary relation” instead of limiting to binary relations between two given sets. This increases abstraction and may increase expressiveness. Particularly this allows to use just “set $X$” instead of “subset $X$ of our object $A$“, that is it allows not to mention the objects for which the sets or filters are considered.