#' is just shorthand for function, just as ' is shorthand for quote.

You can use it anywhere where you want to indicate to the byte-compiler or the interpreter or a human reader that its argument is expected to be (is treated as) a function.

In many contexts the context determines how the argument is treated if, for example, you simply quote it (use quote or ') instead of using #' (or function). For example, in a context where a symbol is used only for its symbol-function property, i.e., it is used as a function, you can just pass the symbol (e.g. by quoting it or passing a variable whose value is the symbol).

But sometimes code is clearer if you do use #' in such contexts.

From (elisp) Anonymous Functions:

-- Special Form: function function-object

This special form returns FUNCTION-OBJECT without evaluating it.

In this, it is similar to quote (*note Quoting::). But unlike quote, it also serves as a note to the Emacs evaluator and byte-compiler that FUNCTION-OBJECT is intended to be used as a function. Assuming FUNCTION-OBJECT is a valid lambda expression, this has two effects:

• When the code is byte-compiled, FUNCTION-OBJECT is compiled into a byte-code function object (*note Byte Compilation::).

• When lexical binding is enabled, FUNCTION-OBJECT is converted into a closure. *Note Closures::.

The read syntax #' is a short-hand for using function. The following forms are all equivalent:

(lambda (x) (* x x))
(function (lambda (x) (* x x)))
#'(lambda (x) (* x x))

In the following example, we define a change-property function that takes a function as its third argument, followed by a double-property function that makes use of change-property by passing it an anonymous function:

(defun change-property (symbol prop function)

   (let ((value (get symbol prop)))
     (put symbol prop (funcall function value))))

(defun double-property (symbol prop)

   (change-property symbol prop (lambda (x) (* 2 x))))

Note that we do not quote the lambda form.

If you compile the above code, the anonymous function is also compiled. This would not happen if, say, you had constructed the anonymous function by quoting it as a list:

(defun double-property (symbol prop)

   (change-property symbol prop '(lambda (x) (* 2 x))))

In that case, the anonymous function is kept as a lambda expression in the compiled code. The byte-compiler cannot assume this list is a function, even though it looks like one, since it does not know that change-property intends to use it as a function.