The pullback lemma
I remember how hard it was to assimilate category theory when I was a student. A beginning student on math.stackexchange.com is asking for a solution to a basic lemma about pullbacks. It really is the sort of thing one should do by oneself. Nevertheless, here it is, in gory details.
Download: pullback.pdf
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The pasting law for pullbacks goes a long, long way when generalized to homotopy (type) theory.
After we had finished the discussion of principal bundle theory "in homotopy type theory" (in abstract higher topos theory) here a colleague came to me and said: I checked all your propositions and I noticed that you are essentially just using the pasting law/pullback lemma over and over and over again. And, yes, that's true. A whole lot folllows this way.