Oftentimes, in the standard algebraic topology books (May, Switzer, Whithead, for instance), there are tricky little proofs that depend on proving that two maps are homotopic. This is comparable to the way we build homotopies, lifts, etc. combinatorially in simplicial homotopy theory, but for some reason I never really acquired the skill-set (maybe the intuition?) to come up with these homotopies in the topological case. I'm just mystified how these little formulas are pulled out of thin air.

Am I missing a key technique that's often taught early-on in an algebraic topology course? Is it tricky even with practice? Have there been any papers that focus on systematic ways of generating these things?

I also noticed that in May's book, he oftentimes writes out explicit formulas for his homotopies, sometimes in a way that obscures the issue at hand (for instance, there is a homotopy that is described by an explicit formula, but it's nothing more than an explicit "representative of the natural homotopy" between the identity map and the constant map on a contractible based space.) How often can these seemingly arbitrary formulas be replaced with more canonical descriptions? (This last question is a soft question to people with experience in topology)