Now this operation “◦” of composition of functions is associative, as follows. If we have a further function h : C → D and form h ◦ g and g ◦ f, then we can compare (h ◦ g) ◦ f and h ◦ (g ◦ f) as indicated in the diagram given above. It turns out that these two functions are always identical, (h ◦ g) ◦ f = h ◦ (g ◦ f ) since for any a ∈ A,wehave ((h ◦ g) ◦ f)(a)=h(g(f(a))) = (h ◦ (g ◦ f))(a)
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LustrzanyDotyk - (no access) - (Oxford Logic Guides 52) Steve Awodey-Category Theory-Oxford University Press (2010).pdf, p20
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