Typescript 可以'；t使用t正确地索引对象；联合；索引类型

首先，我不确定我的代码出了什么问题，所以如果有更好的标题建议，我会编辑它，谢谢.

我试着用联合类型的"key"来索引一个对象，如果键的类型缩小了，没有问题，但如果类型是联合类型，TS就不满意了，为什么？

我也不理解TS说的错误，那个交叉点类型是从哪里来的？

playground个

type FetchedData = {
    A?: {
        A:'A',
        otherField: string
    },
    B?: {
        B:'B',
        otherField: string
    },
    C?: {
        C:'C',
        otherField: string
    },
};

type CachedData = {
    A: {
        A:'A'
    },
    B: {
        B:'B'
    },
    C: {
        C:'C'
    },
};

const cachedData: CachedData = {
    A: {
        A:'A'
    },
    B: {
        B:'B'
    },
    C: {
        C:'C'
    },
}

type Key = keyof FetchedData;

const key: Key = Math.random() > 0.6 ? 'A' : Math.random() > 0.3 ? 'B' : 'C';

const fetchedData: FetchedData = {};

switch (key) {
    case "A" : {
        const cache = cachedData[key]
        fetchedData[key] = {...cache, otherField: 'otherField'}
    }
    // the above is okay, but te below is not, why?
    default : {
        const cache = cachedData[key]
        /* 
            const cache: {
                A: 'A';
            } | {
                B: 'B';
            } | {
                C: 'C';
            }
        */
        fetchedData[key] = {...cache, otherField: 'otherField'}
        /* 
            Type '{ otherField: string; A: "A"; } | { otherField: string; B: "B"; } | { otherField: string; C: "C"; }' is not assignable to type '({ A: "A"; otherField: string; } & { B: "B"; otherField: string; } & { C: "C"; otherField: string; }) | undefined'.
            Type '{ otherField: string; A: "A"; }' is not assignable to type '{ A: "A"; otherField: string; } & { B: "B"; otherField: string; } & { C: "C"; otherField: string; }'.
            Property 'B' is missing in type '{ otherField: string; A: "A"; }' but required in type '{ B: "B"; otherField: string; }'.ts(2322)
        
        */

        // I don't understand why TS says "... not assignable to type '({ A: "A"; otherField: string; } & { B: "B"; otherField: string; } & ..." where does that intersection come from?
    }
}