character, corresponding-character — a character.

If character is a lowercase character, char-upcase returns the corresponding uppercase character. Otherwise, char-upcase just returns the given character.

If character is an uppercase character, char-downcase returns the corresponding lowercase character. Otherwise, char-downcase just returns the given character.

The result only ever differs from character in its code attribute; all implementation-defined attributes are preserved.

 (char-upcase #\a) → #\A 
 (char-upcase #\A) → #\A 
 (char-downcase #\a) → #\a 
 (char-downcase #\A) → #\a 
 (char-upcase #\9) → #\9 
 (char-downcase #\9) → #\9 
 (char-upcase #\@) → #\@ 
 (char-downcase #\@) → #\@ 
 ;; Note that this next example might run for a very long time in 
 ;; some implementations if CHAR-CODE-LIMIT happens to be very large 
 ;; for that implementation. 
 (dotimes (code char-code-limit) 
   (let ((char (code-char code))) 
     (when char 
       (unless (cond ((upper-case-p char) (char= (char-upcase (char-downcase char)) char)) 
                     ((lower-case-p char) (char= (char-downcase (char-upcase char)) char)) 
                     (t (and (char= (char-upcase (char-downcase char)) char) 
                             (char= (char-downcase (char-upcase char)) char)))) 
         (return char))))) 
→ NIL

Should signal an error of type type-error if character is not a character.

upper-case-p, alpha-char-p, Section 13.1.4.3 (Characters With Case), Section 13.1.10 (Documentation of Implementation-Defined Scripts)

If the corresponding-char is different than character, then both the character and the corresponding-char have case.

Since char-equal ignores the case of the characters it compares, the corresponding-character is always the same as character under char-equal.