Oidification
Given some kind algebraic structure, say a gadget, a gadgetoid is a kind of typed or partial gadget, i.e. a category-like structure1 such that a gadget is a single-object gadgetoid. The elements of a gadget become the arrows of a gadgetoid. Thus the process of oidification, also called horizontal categorification, is a kind of section to the process of specializing to a single object.
This is one of two broad approaches for turning something into a category. See also (vertical) Categorification.