The part of the sewer line that the city is responsible for is usually just the main sewer line in the ground but sometimes also the sewer line connected to the main up to the property line.
But in some places even the homeowner is responsible for the sewer main line from the property line all the way till it connects to the sewer main.
The city will dig up the alley or street where the sewer line connects to the sewer main and may install a new pipe from the sewer main to the property line where you can connect your new sewer line into that sewer main drop line that leads into the sewer main.
Other times you may be responsible for connecting to the sewer main yourself or having a plumber do the job.