Connecting a USB to UART converter to a computer's serial DB9 port
If your USB to Serial adapter is one that comes with its own DB-9 connector then that connector supports the typical RS232 voltage levels of the signalling and has connector pin out that matches to the standard serial pinning of a legacy PC serial port. In this case it would be a regular "cross over" cable wiring to connect the two. Alternatively there used to be available DB-9 null modem adapters that contained the cross over internally so you could then use straight wired cables to complete the connection. 

Here is a picture of a typical USB to RS232 adapter of this type. 



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This type of adapter typically has two chips inside. One chip is a USB to UART converter device and the other is a charge pump level translator chip to produce the positive to negative swings of the RS232 signalling. Note that this level translator also inverts the sense of the UART level signals from the USB-UART adapter.

These days with the combination of most PCs and laptops losing RS232 serial ports AND the proliferation of embedded MCU platforms like Arduino and rPi the use of simpler USB to UART adapters has come to be very common. These adapters just have one chip inside which is the USB to UART device and do not include the charge pump level translator chip. Such adapter would not be compatible for connection to a DB-9 RS232 port. Here are some pictures this latter type of adapter.



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