The function shifts the alphabetical characters in the input string by 1 for lowercase and 2 for uppercase letters (cyclically) keeping other characters unchanged.

[CODE]: 
```python
def approx_function(input_string):
    result = ''
    for char in input_string:
        if 'a' <= char <= 'z':
            result += chr(((ord(char) - ord('a') + 1) % 26) + ord('a'))
        elif 'A' <= char <= 'Z':
            result += chr(((ord(char) - ord('A') + 2) % 26) + ord('A')).lower()
        else:
            result += char
    return result
```

To double-check the approximated function's behavior, please run it on some sample inputs:

COMMAND: PYTHON(./temp/function.py "ShiftMeByTwo" "AaBbCc" "!@*&")