Monitors with HDMI connection
The Kensington adapter connects to your MacBook using the USB 3.0 cable.
You will then use a DVI to HDMI cable to connect to the monitor.
Monitors with VGA or DVI connection
For monitors with DVI or VGA connections as shown below:
The Kensington adapter connects to your device using the USB 3.0 cable.
You will then use either a VGA or DVI cable to connect to the monitor. Using VGA will require the use of the VGA to DVI Adapter supplied in the Kensington box.
You should continue to use the method you have been using to connect to your projector. For example, Chromebooks connect using the USB-C to HDMI/VGA adapter to the projector. Macbooks connect with Mini display port to VGA or HDMI depending on your projector connection type. This will allow to be connected to both the monitor and the projector simultaneously.