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I'm specifically wondering about the spells Armor of Agathys, Fire Shield, and Shield, but a general answer would be helpful. For example, if a level 9 Abjuration Wizard/level 2 Warlock multiclass with an Arcane Ward with 20 hit points active, Fire Shield cast at 4th level, and Armor of Agathys cast at 5th level is hit by a melee attack that deals 15 damage, would the attacker take 25 points of cold damage from Armor of Agathys and 2d8 points of fire damage from Fire Shield, even though the attack did damage to your Arcane Ward, and not you?

Could you cast Shield in response to the attack in the example above to try and make it miss, or would that not work because Shield must be cast as a reaction to you being hit by an attack and the attack hit the Arcane Ward, instead of you?

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You are still hit by the attack so yes, spell effects the trigger on a hit would still trigger.

From page 194 of the PHB:

Making An Attack

Whether you're striking with a melee weapon, firing a weapon at range, or making an attack roll as part of a spell, an attack has a simple structure.

[...]

3. Resolve the attack. You make an attack roll. On a hit, you roll damage...

And Arcane Ward states:

Whenever you take damage, the ward takes the damage instead. If this damage reduces the ward to 0 hit points, you take any remaining damage.

Damage is rolled after the attack hits and the attack roll and the hit are made against you, not the Arcane Ward. Nowhere in Arcane Ward's description does it modifier you being hit with an attack, it only modifies the damage you take after the hit.

Therefore, spells such as Armor of Agathys, Fire Shield, and Shield would still trigger.

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