> just a block cipher in the category of endofunctors

A block cipher is just a keyed pseudorandom permutation! :)

Imagine that we have arranged all numbers from 0 to 115792089237316195423570985008687907853269984665640564039457584007913129639936 (2^256-1) in an array and then shuffled that array 115792089237316195423570985008687907853269984665640564039457584007913129639936 (2^256) times uniquely, each time recording the resulting shuffled array.

Here's the Go-like type:

var perm [115792089237316195423570985008687907853269984665640564039457584007913129639936]uint256

This is an unkeyed permutation (we can already build a secure hash function like SHA-3 from it, e.g. SHA-3 uses [2^1600]uint1600 permutation, while Ascon-Hash uses [2^320]uint320. The best we can do with ours is 2^127.5 collision security).

A keyed permutation takes 2^256 of these arrays, again shuffled differently and uniquely, so we have:

var keyedPerm [115792089237316195423570985008687907853269984665640564039457584007913129639936][115792089237316195423570985008687907853269984665640564039457584007913129639936]uint256

A block cipher is just P[k][p] where k and p are indexes into the arrays. Let's call k the key and p the plaintext — first we select one particular permutation using the key, and then select the resulting number (ciphertext) using the plaintext.

A simple hash function built from this keyed permutation splits the message into 256-bit numbers and then selects permutations using the message as k, and previous value as p and adds them together:

   h = (some starting number called IV)
   h += keyedPerm[m[0]][h]
   h += keyedPerm[m[1]][h]

This is SHA-2, SHA-1, MD5, etc, and with a slight variation and a larger block cipher, BLAKE(1,2,3).

Of course, it's physically impossible to have that much memory for the arrays and it's physically impossible to shuffle that many times, so a block cipher is an approximation of that by smashing and rearranging bits, hoping to cause diffusion and confusion.

Chopped, Stored, Secured – The Story of the Hash Function