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We create detailed mock catalogues of AGN spectra from the rest-frame near-infrared down to the ultraviolet \u2013 including emission lines \u2013 to simulate what<jats:italic>Euclid<\/jats:italic>will observe for both obscured (type 2) and unobscured (type 1) AGN. We concentrate on the red grisms of the NISP instrument, which will be used for the wide-field survey, opening a new window for spectroscopic AGN studies in the near-infrared. We quantify the efficiency in the redshift determination as well as in retrieving the emission line flux of the H<jats:italic>\u03b1<\/jats:italic>+[N\u202f<jats:sc>II<\/jats:sc>] complex, as<jats:italic>Euclid<\/jats:italic>is mainly focused on this emission line, given that it is expected to be the brightest one in the probed redshift range. Spectroscopic redshifts are measured for 83% of the simulated AGN in the interval where the H<jats:italic>\u03b1<\/jats:italic>is visible (i.e. 0.89\u2004&lt; \u2004<jats:italic>z<\/jats:italic>\u2004&lt; \u20041.83 at a line flux of &gt; 2\u2005\u00d7\u200510<jats:sup>\u221216<\/jats:sup>erg s<jats:sup>\u22121<\/jats:sup>cm<jats:sup>\u22122<\/jats:sup>, encompassing the peak of AGN activity at<jats:italic>z<\/jats:italic>\u2004\u2243\u20041\u2005\u2212\u20051.5) within the spectral coverage of the red grism. Outside this redshift range, the measurement efficiency decreases significantly. Overall, a spectroscopic redshift iscorrectly determined for about 90% of type 2 AGN down to an emission line flux of roughly 3\u2005\u00d7\u200510<jats:sup>\u221216<\/jats:sup>erg s<jats:sup>\u22121<\/jats:sup>cm<jats:sup>\u22122<\/jats:sup>, and for type 1 AGN down to 8.5\u2005\u00d7\u200510<jats:sup>\u221216<\/jats:sup>erg s<jats:sup>\u22121<\/jats:sup>cm<jats:sup>\u22122<\/jats:sup>. Recovered values for black hole mass show a small offset with respect to the input values by about 10%, but the agreement is good overall. With such a high spectroscopic coverage at<jats:italic>z<\/jats:italic>\u2004&lt; \u20042, we will be able to measure AGN demography, scaling relations, and clustering from the epoch of the peak of AGN activity down to the present-day Universe for hundreds of thousands of AGN with homogeneous spectroscopic information.<\/jats:p>","DOI":"10.1051\/0004-6361\/202348326","type":"journal-article","created":{"date-parts":[[2024,2,12]],"date-time":"2024-02-12T19:50:16Z","timestamp":1707767416000},"page":"A108","source":"Crossref","is-referenced-by-count":7,"title":["<i>Euclid<\/i>preparation"],"prefix":"10.1051","volume":"685","author":[{"name":"Euclid 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