The function processes input strings. If a character is a lowercase letter, it increments the letter by 2 positions in the alphabet, wrapping back to the beginning if necessary (e.g. 'y' becomes 'a', 'z' becomes 'b'). If the character is an uppercase letter, it increments by 1 position with wrap-around (e.g. 'Z' becomes 'A'). All other characters remain unchanged.

[CODE]: Here's a Python code that approximates the behavior of the function:

```python
def approximate_function(input_string):
    output = ""
    for char in input_string:
        if 'a' <= char <= 'z':
            output += chr(((ord(char) - ord('a') + 2) % 26) + ord('a'))
        elif 'A' <= char <= 'Z':
            output += chr(((ord(char) - ord('A') + 1) % 26) + ord('A'))
        else:
            output += char
    return output
```

Please note that this is just an approximation of the function behavior, and there may be minor discrepancies in some edge cases.