Stop pretending studying arrow theory improves your apping

Given two functors S,T:C→B, a natural transformation τ:S→T is a function which assigns to each object c of C an arrow τ_c=τc:Sc→TC of B in such a way that every arrow f:c→c' in C yields a diagram
ﾠﾠﾠﾠﾠﾠﾠﾠﾠﾠﾠτc
c ﾠﾠ Sc---→Tc
|ﾠﾠﾠﾠﾠﾠﾠﾠ|ﾠﾠﾠﾠﾠﾠﾠ|
|fﾠﾠﾠﾠﾠSf↓ﾠﾠﾠﾠﾠ↓Tf
↓ﾠﾠﾠﾠﾠﾠﾠ|ﾠﾠτc'ﾠﾠ|
c', ﾠﾠﾠ Sc'--→Tc'

which is commutative. When this holds, we also say that τ_c=τc:Sc→Tc is natural in c. If we think of the functor S as giving a picture in B of (all the objects and arrows of) C, then a natural transformation τ is the set of arrows mapping (or, translating) the picture S to the picture T, with all squares (and parallelograms!) like that above commutative:
ﾠaﾠﾠﾠﾠﾠﾠﾠﾠﾠﾠSa----------→Ta
ﾠ|ﾠ╲fﾠﾠﾠﾠﾠﾠﾠ ﾠ|ﾠ╲Sfﾠﾠﾠﾠﾠﾠﾠﾠﾠ|ﾠ╲Tf
ﾠ|ﾠﾠﾠ↘ﾠﾠﾠﾠﾠﾠﾠ|ﾠﾠﾠ↘ﾠﾠﾠτbﾠﾠﾠ|ﾠﾠﾠ↘
ﾠ|ﾠﾠﾠﾠﾠbﾠﾠﾠﾠ ﾠ|ﾠﾠﾠﾠSb----------→Tb
ﾠ|ﾠﾠﾠ╱ﾠﾠﾠﾠﾠﾠﾠ|ﾠﾠﾠ╱ﾠﾠﾠﾠﾠﾠﾠﾠﾠ|ﾠﾠﾠ╱
↓ﾠ↙ﾠﾠﾠﾠﾠﾠ ﾠ↓↙Sgﾠﾠﾠﾠﾠﾠﾠﾠ↓ﾠ↙Tg
ﾠcﾠﾠﾠﾠﾠﾠﾠﾠﾠﾠSc----------→Tc

>>13
You don't seem to be getting it.. there's math that is relevant and useful to a given problem and math that isn't.

There is mathematics generalized enough that the very basic definitions from it can be applied to anything: because it's "math" most people are too scared to question it but it's basically snake oil - it is completely useless for programming.

People have been writing physics engines for years using linear algebra and numerics. Trying to shoehorn yonedas lemma, topos theory and comonadics in doesn't improve the software. It's pointless they just do it to sound smart.