This might be more of a mathoverflow question, but here it goes...
For a number field $K$, Marcolli in http://www.its.caltech.edu/~matilde/QSManabelian.pdf page 10 calls $G^\text{ab}_K \times_{\hat{\mathcal{O}}^*_K} \hat{\mathcal{O}}_K$ a "fibered product".
But her description of this space sounds more like the direct product $G^\text{ab}_K \times \hat{\mathcal{O}}_K$ quotiented by the action of $\hat{\mathcal{O}}^*_K$ by the Artin map, so that the points of this space are $\hat{\mathcal{O}}^*_K$-orbits.
The fibered product I know is described at https://en.wikipedia.org/wiki/Pullback_(category_theory) ? Can these two be reconciled? The latter fibered product is strictly a subset of the direct product. The best I can hope for is that this subset is a complete set of distinct representatives of the orbits of the action. But I cannot figure out what the underlying maps $G^\text{ab}_K,\hat{\mathcal{O}}_K \to \hat{\mathcal{O}}^*_K$ would be.