Context. Unipolar magnetic regions are often associated with supergranular cells. The chromosphere above these regions is regulated by the magnetic field, but the field structure is poorly known. In unipolar regions, the fibrillar arrangement does not always coincide with magnetic field lines, and polarimetric observations are needed to establish the chromospheric magnetic topology. Aims. In an active region close to the limb, we observed a unipolar annular network of supergranular size. This supergranular structure harbours a radial distribution of the fibrils converging towards its centre. We aim to improve the description of this structure by determining the magnetic field configuration and the line-of-sight velocity distribution in both the photosphere and the chromosphere. Methods. We observed the supergranular structure at different heights by taking data in the Fe i 6301-6302 Å, Hα, Ca ii 8542 Å and the Ca ii H&K spectral lines with the CRISP and CHROMIS instruments at the Swedish 1-m Solar Telescope. We performed Milne-Eddington inversions of the spectropolarimetric data of Fe i 6301-6302 Å and applied the weak field approximation to Ca ii 8542 Å data to retrieve the magnetic field in the photosphere and chromosphere. We used photospheric magnetograms of CRISP, HINODE/SP and HMI to calculate the magnetic flux. We investigated the velocity distribution using the line-of-sight velocities computed from the Milne-Eddington inversion and from Doppler shift of the K 3 feature in the Ca ii K spectral line. To describe the typical spectral profiles characterising the chromosphere above the inner region of the supergranular structure, we performed a K-mean clustering of the spectra in Ca ii K. Results. The photospheric magnetic flux shows that the supergranular boundary has an excess of positive polarity and the whole structure is not balanced. The magnetic field vector at chromospheric heights, retrieved by the weak field approximation, indicates that the field lines within the supegranular cell tend to point inwards, and might form a canopy above the unipolar region. In the centre of the supergranular cell hosting the unipolar region, we observe a persistent chromospheric brightening coinciding with a strong gradient in the line-of-sight velocity.