We think of the composition g ◦ f as a sort of “product” of the functions f and g, and consider abstract “algebras” of the sort arising from collections of functions. A category is just such an “algebra,” consisting of objects A,B,C,... and arrows f : A → B, g : B → C, ..., that are closed under composition and satisfy certain conditions typical of the composition of functions.
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LustrzanyDotyk - (no access) - (Oxford Logic Guides 52) Steve Awodey-Category Theory-Oxford University Press (2010).pdf, p18
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