Replacing Lamp Holders

I have removed 2 motion sensor lamp holders from daughter’s basement ceiling, trying to install a basic porcelain Leviton lamp holder(model 49875-660W/250v) that has double screw terminals(4 total) on the back. My limited knowledge of electrical only extends to changing/installing light switches and/or lamp holders & chandeliers many times over the years; this is the first time I’ve run into two black wires and two white wires in junction box; typically just one each. There are two lamp holders in her basement, about 8-10 feet apart in direct line of each other. I’m just trying to give her light anytime she turns on the light switch instead of only when motion detected and then the lamp shuts off.

The black & white wires in junction box are pretty thick, maybe 3x the thickness of the 16 or 18 awg(hard to read) that was on the motion detector lamp holder.

What I’m wondering is, if I can cut a black and white wire from the motion detector and connect it to the black and white in the junction box with wire caps, then connect that 16 or 18 awg to my Leviton lamp holder.

Thanks so much,

Don