Endorsement is an item in the “Plot advancement” category (Key Items).
In-game descriptions
| Sword / Shield | A letter of recommendation that you’ll need in order to participate in the Gym Challenge. |
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Where to find it
Not placed on the atlas maps (so far: Red/Blue and Generation 3).
Sources : PokeAPI (items, item_flavor_text, machines) · atlas · pokeemerald species_info.h