Last year I asked a question titled Direct comparison zig-zag between cochain theories on MathOverflow, and it has still no good answer. I’ve offered a bounty, and no takers yet. The satisfaction is yours for the taking! To quote from the end of the question:
…my hope is to show that the class of cochain theories is connected, assuming one exists, and then exhibit one. This may well be singular cohomology, or it might be something else. In particular I want to remove from the proof of uniqueness the privileged position that any one construction has. The only caveat is that I won’t be able to use any super-sophisticated machinery as this is a first course in algebraic topology. I’m happy to have an outline of how to unwind a sophisticated proof.