Do I need to use remote events to change the cframe of a local part?

I have a script which attempts to set the cframe of a local part to the cframe of a part in the workspace, so that upon tweening, the camera's position is set the local parts position.

The entire objective here is to get the camera to tween from where it stops so any alternatives are appreciated!

All of the variables are defined and work. The function is being called and the camera tweening is occurring only it starts from the local parts original position rather than moving the local part to T4's position.

--CT is the local part
function RoomCam()
camera.CoordinateFrame = CFrame.new(T3.Position)
camera.Focus = T4.CFrame
camera:Interpolate(T4.CFrame, T4.CFrame + T4.CFrame.lookVector, 3)
CT.Position = T4.CFrame

end