Natural Transformation

Upon opening the puzzle, we see sets of diagrams, each with images and circles, linked together by arrows. Different arrows have different icons. The first set of circles show images for all of the circles, each of which can be identified.

We then can use these clues to infer the meaning of the the arrows and their icons:

Icon	Meaning	Example
	Derivative (not just mathematical definition)	Nut → Nut butter
	Substring	perm → permit
	Adding co- to start of word	bra → cobra
	x 1000	kilo → mega
	Element of/type of	March → Months
	Concatenate	F + B → FB
	Abbreviate	Facebook → FB
	NATO letters	N → November
	Element symbol	Boron → B
	Animal collective noun	cobra → quiver
	Inverse (not just mathematical definition)	sqrt(x) ↔ x2

Upon determining the meaning of the arrow symbols, we can solve the later diagrams. It is also possible to use the later diagrams to determine the meaning of the arrow symbols that may not be as clear to obtain from the first diagram.

Upon solving all 6 subparts, we can solve the extraction diagram to arrive at a circle containing -log(cos(gram)). The derivative of this is tan(gram), giving the final answer TANGRAM.

Authors' Notes

This puzzle was inspired by category theory, hence the title of the puzzle and the co-arrow. In fact, the clue that was the spark for the puzzle was the cluing of coconut → conut → nut → nut butter. However, it ended up being hard to find more dumb math jokes we could include, and for the sake of making the puzzle more accessible, we removed the category theory notion of co- meaning duals.

I still don’t like category theory.

Appendix: Solved Diagrams