When reading various papers about the lambda calculus, ISWIM and a number of other things, I have heard the word "postcompose" come up a lot (e.g. in https://en.m.wikipedia.org/wiki/J_operator). However, after a lot of research, I could not find anything (except one definition that mathematical and unrelated to what I was looking for). So, what does "postcompose" mean?
The term postcompose is not specific to the lambda calculus. It really just means compose in the context of two functions, operators, functors or whatever objects on which composition is defined. The post prefix is used to disambiguate the order of composition, so postcomposing f with g is g ∘ f = g(f(x)).
In the context of lambda calculus, you have e.g. function application which is composable.