My friend left the adapter for her Nintendo Switch at a friend's house. She is still able to charge the console and play it in handheld mode. Yet, she is unable to connect it to her T.V.. It seems that, being that she can successfully charge her switch, she -should- be able to connect it to her T.V. given the HDMI cable provided by Nintendo. Why can't she connect it to her T.V.?
It would seem that the official Nintendo AC adapter for the Switch is significantly more powerful than other USB-C chargers, providing up to 39W whereas most USB-C chargers apparently provide less. (I'm having some difficulty finding precise numbers for the average USB-C charger at the moment.)
The Switch itself only needs 9.1W to charge, according to the linked site, therefore any old USB-C charger can accomplish the task of charging a Switch, however, the docking station also has 3 USB-A ports which could have peripherals attached, for all the system knows. In order to avoid a situation where all the things (potentially) connected to the dock are drawing more power than the charger can provide, some people are theorizing that Nintendo added logic in the dock's interchange that prevents anything more than charging to occur with AC adapters other than their own connected.
This does seem to be consistent with your friend's observations, as well as my own - I have noticed that my Switch won't pipe video to the TV if the AC adapter is unplugged. Even when the Switch itself is fully charged and should, in theory, be able to run for at least a couple of hours without power input anyway.
(It's the only way to be sure the connected AC Adapter is providing enough power, since theirs is designed to provide more than most USB-C chargers will.)