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The topos of functions Sets^(->), e.g my theory, has as its objects functions as its morphisms m: f -> g function pairs (h,k) that make a commutative square with gh = kf. If we take function pairs like (h,k) as our morphisms then we can do several things to simplify them: like making the two functions in the pair equal or making one of them into an identity.
1. precomposition: with precomposition you should use a function pair (h, 1_b) that has an identity for the second part to form a morphism m: fg -> f like so: https://q.uiver.app/?q=WzAsNCxbMCwyLCJBIl0sWzIsMiwiQiJdLFswL...
2. postcomposition: with composition you should use a function pair (1_b, k) that has an identity for the first part to form a morphism m: f -> hf like so: https://q.uiver.app/?q=WzAsNCxbMCwyLCJBIl0sWzIsMiwiQiJdLFsyL...
3. conjugation: with conjugation you can actually use an equal function pair (g,g) to morphism a function f to its conjugate gfg^{-1}. Then we see that this forms a commutative square for gfg^{-1}g = gf. Thusly, there is always a morphism from any function to its conjugate by a bijection like so: https://q.uiver.app/?q=WzAsNCxbMCwyLCJBIl0sWzIsMiwiQSJdLFswL...
Therefore, I would say that precomposition, postcomposition, and conjugation form special types of morphisms between functions defined by special properties like having one function be the identity or having both functions be equal to one another. Then beyond that we can classify these morphisms by rather or not their components are projections, injections, or bijections to get towards what you are speaking about. This all fits in to the perspective that functions are objects in Sets^(->), so that is nice.
I can't say I've ever used APL but I have heard of it. It has some very interesting syntax and notation, but I have made it far enough to actually try it out yet.