HDMI-to-SDI conversion on output from recorders

For the Video Assist 4K and the HyperDeck Shuttle 2, I'm wondering if I can use HDMI (1080i59.94, if it matters) for input and then output video via SDI?

To be clear: I don't want to convert the resolution/framerate. I just want to record video via the HDMI input, and output an identical video signal over SDI.

If so:

1. Does it require any special setup/configuration?

2. If the answer is yes for the Video Assist 4K: Does the outputted video embed the audio from the device's XLR inputs, assuming I've set that to be the audio source?

Yes, the HDMI input on the VA and Shuttle 2 will be output on the SDI out, but there is a s,Ishtar delay, as the signal is processed in the Devices and re-output from the recording processor. Not sure about direct audio in on the VA4K, but it seems it should embed the audio from the inputmsnd output in the SDI out.
Cheers