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So, this is not much of a question, more like notes on what I’m currently working on. If you find something unclear in my description, it is probably because I have made some mistake somewhere, so please don't hesitate to point that out. If you have some paper or book in mind that might be beneficial for clarifying, proving or formalizing my concepts, I am more than interested.

So, this is not much of a question, more like notes on what I’m currently working on. If you find something unclear in my description, it is probably because I have made some mistake somewhere, so please don't hesitate to point that out. If you have some paper or book in mind that might be beneficial for clarifying, proving, disproving or formalizing my concepts, I am more than interested.

I call it parallel because Milewski has taught me to imagine functors as pointing out parallel planes in the domain category, and natural transformations as arrows for going from one plane to the other. My parallel transformation cannot do this, but if you compose the hexagon you get an arrow

$$ \operatorname{hom}(\A, \B) \to \operatorname{hom}(||\sigma_1|| (\B; \A) \to ||\sigma_2|| (\A; \B) $$

I call it parallel because Milewski has taught me to imagine functors as pointing out parallel planes in the domain category, and natural transformations as perpendicular arrows going from the first plane to the second. My parallel transformation cannot do this, but if you compose the hexagon you get an arrow

$$ \operatorname{hom}(\A, \B) \to \operatorname{hom}(||\sigma_1|| (\B; \A), ||\sigma_2|| (\A; \B)) $$