てもいい isn't permission — it's a sign that the door is open

Where the picture shows up in てもいい

1. 1

   asking if the door is open

   すみません、ここに座ってもいいですか。

   sumimasen, koko ni suwatte mo ii desu ka

   Excuse me, may I sit here?

   More polite: 座ってもよろしいでしょうか.

2. 2

   opening the door for someone

   体調が悪いなら、今日は早く帰ってもいいですよ。

   taichō ga warui nara, kyō wa hayaku kaette mo ii desu yo

   If you're not feeling well, you can leave early today.

3. 3

   a door in front of a noun

   支払いは現金でもいいですか。

   shiharai wa genkin de mo ii desu ka

   Is it OK if I pay in cash?

   Nouns take で: 現金でもいい.

4. 4

   a door you needn't walk through

   無理して全部やらなくてもいいですよ。

   muri shite zenbu yaranakute mo ii desu yo

   You don't have to push yourself and finish everything.

   なくてもいい is not a ban — it's the absence of obligation. Do it or don't.

5. 5

   offering to walk through yourself

   よかったら、私が代わりに行ってもいいですけど。

   yokattara, watashi ga kawari ni itte mo ii desu kedo

   If you like, I could go instead.

   行きます settles it; 行ってもいいですけど leaves the other person room to decline.