I've read through one class field theory text after another, but there's something very non-intuitive for me about cohomology that makes it hard for me to understand why Tate cohomology was invented.

In order to make up for my lack of intuition regarding anything cohomological, I have gone through a few texts introducing the derived category framework. This approach has been very helpful for me, and has helped me gain intuition about cohomological methods in Algebraic Topology and Algebraic Geometry. My hope is that, with your help, it could also shed light on "Tate cohomology", which, at the moment, seems to me like a completely arbitrary definition that helps in mysterious and miraculous ways.

Is there a definition of Tate cohomology via derived categories that sheds light on why they are a natural thing to consider? What would be a reference of such a treatment? (If you think that I am looking in the wrong place for intuition and you have an alternative suggestion, I would be very happy to hear that as well!)