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Instruction On Eucharistic Worship issued by the Sacred Congregation of Rites
While the Blessed Sacrament is exposed, the celebration of Mass in the same area of the church (eadem aula ecclesiae) is forbidden .... This is because ... the celebration of the Mystery of the Eucharist includes in a more perfect way that spiritual communion to which exposition should lead the faithful. Therefore there is no need for this further help. If exposition of the Blessed Sacrament is prolonged for a day, or for several successive days, it should be interrupted during the celebration of the Mass, unless it is celebrated in a chapel apart from the exposition area and at least some of the faithful remain in adoration.
Code of Canon Law, canon 943:
"... In special circumstances the minister of exposition and deposition alone, but without the blessing, is an acolyte, an extraordinary minister of holy communion, or another person deputed by the local Ordinary, in accordance with the regulations of the diocesan Bishop."
Instruction Redemptionis Sacramentum
"[156.] This function is to be understood strictly according to the name by which it is known, that is to say, that of extraordinary minister of Holy Communion, and not “special minister of Holy Communion” nor “extraordinary minister of the Eucharist” nor “special minister of the Eucharist”, by which names the meaning of this function is unnecessarily and improperly broadened."
"... In special circumstances the minister of exposition and deposition alone, but without the blessing, is an acolyte, an extraordinary minister of holy communion, or another person deputed by the local Ordinary, in accordance with the regulations of the diocesan Bishop."
Instruction Redemptionis Sacramentum
"[156.] This function is to be understood strictly according to the name by which it is known, that is to say, that of extraordinary minister of Holy Communion, and not “special minister of Holy Communion” nor “extraordinary minister of the Eucharist” nor “special minister of the Eucharist”, by which names the meaning of this function is unnecessarily and improperly broadened."
In other words, no Eucharistic Ministers may set up Adoration unless there are special circumstances and with permission of the local bishop. . Only priests, deacons, seminarians who are acolytes and members of orders as indicated in the above document. Extra-ordinary Ministers of Communion are not the Ordinary Ministers of Communion.
The various correct titles are Ordinary and Extraordinary Ministers, not Eucharist Minister.