Adding two tensors in different coordinate systems

I have difficulties understanding the following (Everything out of the book Gravity by Poisson and Clifford): We have a quantity $mathfrak{g}^{alpha beta}$ called gothic inverse metric in prior unspecified coordinates. Then we impose the four harmonic coordinate conditions $partial_{beta}mathfrak{g}^{alpha beta}=0$. So far so good. Now they define the potentials $$ h^{alpha beta} := eta^{alpha beta} – mathfrak{g}^{alpha beta} ,,$$
where $eta^{alpha beta}:= diag(−1, 1, 1, 1)$ is the Minkowski metric expressed in Lorentzian coordinates. I don’t understand in what coordinates the quantity $h^{alpha beta}$ is written now. Maybe this is a stupid question and I just don’t understand it, but what does it even mean do add the fields $eta$ and $mathfrak{g}$ when they are written in different sets of coordinates.
Any help is greatly appreciated!