Q:

How do I customize the way Emacs prints out certain keys and key sequences?

For example, I want Control key to be displayed not as C but as ⎈ (U+2388 HELM SYMBOL) in the minibuffer.
Trying to personalize my Emacs and display all function keys as ISO symbols. And sequences to be displayed as something like ⎈ + x, ⎈ + f.
Is there a way to do it?

A:

Yes. There is a function key-description that takes a list or vector of keys and returns a string that describes them. This is used by the built-in help facilities such as describe-key and describe-function to display information about key bindings. It calls single-key-description on each element of the input list. Both of these are written in C rather than Lisp, but you could in principle override both of these to do what you want by adding advice to them.
key-description is even called by the Info system to render the correct key bindings in the documentation, even once the user starts customizing them.

