The fork() function isn't only to copy the father process, it returns a value which refer that the process is the father or the son process, the image below explain how can you make use of fork() as a father and a son:
as shown when the process is the father fork() returns the son process ID PID else it returns 0
for example you can make use of it if you have a process ( web server ) that receive the requests and on each request it create a son process to process this request, here the father and its sons have different jobs.
SO, no run a copy of a process isn't the exact thing as fork().