I published online my definitions of basic category theory terms (category, precategory, functor, prefunctor, morphism, etc.) with some simple theorems about them. My definitions differ of the standard encyclopedic definitions. I suggest to use my definitions for any serious math research instead of the standard definitions. (I use my definitions for http://www.mathematics21.org/category-theory.html">my category theory research.)
Additionally, I have elegantly defined a category based on a semigroup.
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