Hello Themis!
The entire job is performed by the home hub:
To check if the home hub(s) work(s) as expected, please open the app Home from Apple and tap on the tiny home icon in the top left. Goto
Home Settings > Hubs & Bridges
One home hub must be listed at Home Hubs marked Connected. The others must be marked as Standby.

Basics: In the app Eve you have set
Settings > Accessories > Eve Outdoor Cam > Motion Light, you have set an independent value that comes into effect for the Motion Light function when you use the Stream & Allow Recording feature.

Eve always displays the floodlight - as an element in the Eve Outdoor Cam detail view - as a separate device. If you want the same convenience in the Home app, open the camera settings in Home and tap the "Show as separate fields" option. The floodlight now appears as a tile accessories. Tap and hold the tile and then tap on the small gear. In the field on top assign a name, I chose Flood Light.
Tap the tile to check the functionality.

Now create an automation, when motion is detected by the sensor integrated in Eve Outdoor Cam, the floodlight turns on and turns off after one minute (for example) from the time when no more motion is detected.

In the Home app, select Automation, tap the small + sign on top, and then Add Automation.
Select "A Sensor Detects Something"
Scroll to the room with Eve Outdoor Cam and activate Eve Outdoor Cam, tap Next.
Leave all settings as suggested initially and tap Next.
Scroll down to the room with Eve Outdoor Cam and tap it, in my case Flood Light.

Tap Next. In the overview, tap the tile with Flood Light so it says Turn On. Tap Disable at the bottom and select After 1 min.
Tap Done and test.

In case the home hub(s) work as expected, it will work.

Please keep in mind: In general, the day for HomeKit starts at 00:00 (12:00 am) and ends at 23:59 (11:59 pm). HomeKit does not 'know' the days before and after. So, for example, the condition 'after 23:00 (11 pm)' AND 'before 06:30 (am)' or ‚At Night’ can never be met on the same day.
Here you need to create two rules/automations. One with the condition from 23:00 to 23:59 (11:00-11:59 pm) and one from 00:00 (12:00 am) to 06:30 (am).
Aug 8, 2022