Relinquished's equipped monster is destroyed if Relinquished's ATK is lower than the foe's attacking monster's ATK. If Relinquished is in Defense position, its equipped monster is destroyed if its DEF is less than the attacking monster's ATK. Either way, Relinquished remains on the field.
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It is either a monster that can function as an Equip card through its own effect, such as Union monsters, or, it is a monster forcibly equipped by another card's effect such as Relinquished's ignition effect.
Relinquished is Ritual Summoned with the Ritual Spell Card Black Illusion Ritual. It requires a monster as a Tribute to Special Summon Relinquished from your Hand. Once Relinquished is successfully Ritual Summoned, if it is sent to the Graveyard, it is eligible to be Special Summoned from it. You cannot Special Summon Relinquished from the Graveyard unless it was successfully Ritual Summoned first.
As an Ignition Effect monster, Relinquished can only take a monster in your own turns, in your own main phases. Monsters who can activate in the opponent's turn will usually specifically state so, or be a trigger which activates when a certain condition is met, regardless of turn.
"Relinquished" + "Thousand-Eyes Idol" As long as this card remains face-up on the field, other monsters cannot change their battle positions or attack. Select 1 monster on your opponent's side of the field, and equip it to this card (this effect can only be used once per turn, and you can only equip 1 monster at a time to this card). The ATK and DEF of this card become the same amounts as the monster equipped to this card. If this card is destroyed as a result of battle, the equipped monster is destroyed instead.
Monsters who go into the S/T zone for any reason are then treated as Continuous Spell Cards or Equip Spell Cards depending on what it is doing. It is no longer a monster card so is unaffected by anything that specifically affects monster cards, and cannot be tributed for a tribute summon, etc. However if something required you to send an Equip Spell Card to the graveyard, you could use that monster.