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I'm looking to understand the conceptual process that brought Eilenberg and Mac Lane in developing the basic concepts of category theory.

I quote Mac Lane's book "Category theory for working mathematicians":

"...An adequate treatment of the natural isomorphisms occurring for such limits was a major motivation of the first Eilenberg-Mac Lane paper on category theory [The general theory of natural equivalence]..."

Here's my question:

What are these natural isomorphisms that Mac Lane were referring to?

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You could look at the original paper. There you'll find an extensive discussion and many applications to group theory and topology. – t.b. Nov 23 '11 at 9:23
    
@t.b.: thanks. I've already read that paper but it doesn't explacoverin the path that guided the authors to the creations of those categorical concepts, that is the mathematical philosophy behind category theory. – Giorgio Mossa Nov 23 '11 at 10:08
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Okay, I see. Then I think you could do worse than look at Chapter 2 of Krömer's Tool and Object: A History and Philosophy of Category Theory, here's the table of contents of that chapter: part 1, part 2 and here's an extensive review by Corry of the entire book. – t.b. Nov 23 '11 at 10:52
    
"I didn't invent categories to study functors; I invented them to study natural transformations." - Mac Lane Just some context. – PyRulez Feb 21 '15 at 22:42

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