Database
[{"_id":1,"Name":"HelloWorld","OtherFields":"OtherValues","Projects":[{"Project":{"key":111},"Category":[{"No":"123"},{"No":"987"}]},{"Project":{"key":222},"Category":[{"No":"123"},{"No":"987"}]}]},{"_id":2,"Name":"HelloWorld","OtherFields":"OtherValues","Projects":[{"Project":{"key":111},"Category":[{"No":"123"},{"No":"987"}]},{"Project":{"key":222},"Category":[{"No":"123"},{"No":"987"}]}]}]
Query
db.collection.update({"_id":1,"Projects.Project.key":111},{"$pull":{"Projects.$.Category":{"No":"123"}}})