chris rogers and i tried to figure out a nice definition of "abelian category" that makes the "distributivity" nature of the compatibility relation more manifest. we decided that "the cokernel of the kernel is the kernel of the cokernel, via the invertibility of the natural map from the one tpo the other" probably works, and seems pretty nice. so i wonder whether that's a standard alternative approach.

??perhaps "kernel of cokernel" deserves to be called "image", and "cokernel of kernel" "coimage" ?? ... or something like that...

??so _is_ there some adjointness relationship between kernel and cokernel here??

hmm, i seem to be getting now that cokernel really is just the left adjoint of kernel, and that this is an even nicer way of expressing the compatibility relation... well, nicer in certain ways at least... for example as a precursor to the "stable" world where homotopy cokernel is the inverse of homotopy kernel...

??feels like the adjointness has something to do with "snake lemma" ??? or something??

well, wait a minute... how would the adjointness imply the "commutativity"??? hmmm...

i was also going to mention that the fact that the distributivity natural transformation here seems invertible (is that correct??) makes the analogy to a braiding a bit closer... ??is that in fact showing up here in some way now??

some confusion here... adjoint endofunctors don't necessarily "commute" with each other, right??

[x X y]^y vs [x^y] X y ...??...

inverse ones do, though... ??? ...???...