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camera pose rotation to local rotation for an object

Suppose I have the pose of an object in a 4 x 4 matrix (has translation and rotation information) but it's with respect to my camera frame Suppose I also have the intrinsic calibration off the camera that was used to find the pose of the object

Now I want to get the object rotation but in it's local coordinate system not in the camera frame coordinate system

What is the math to perform this transformation?

I have a feeling this is pretty easy to do but I just can't get my head around it. Is it something like the inverse of the camera intrinsic multiplied by 4 x 4 pose information?