てもいい
temoii
てもいい isn't permission — it's a sign that the door is open
The picture The door in front of you is standing open. Walk through and nothing bad happens. てもいい is the door standing open; てはいけない is a bar dropped across that same door.
Where the picture shows up in てもいい
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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?
You give the door in front of the empty chair a light push to see whether it is open.
More polite: 座ってもよろしいでしょうか.
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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.
Your boss pushes the door to the way home open for you and waves you through.
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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?
At the till, a card door and a cash door stand side by side, both of them open.
Nouns take で: 現金でもいい.
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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.
The door is open, but walking straight past it changes nothing at all.
なくてもいい is not a ban — it's the absence of obligation. Do it or don't.
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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.
You rest one foot in the open doorway and wait for the other person to nod.
行きます settles it; 行ってもいいですけど leaves the other person room to decline.
All of it in one picture
てもいい only ever does one thing: it tells you a door is open. Push it to see whether it gives and you get a question; push it open for someone else and you get permission. In front of a noun, で slips in and another door opens — 現金でもいい — while なくてもいい is a door that stands open but need not be walked through. Rest your own foot in the doorway and it becomes a careful offer. Rather than memorizing 'you may', just look at which door is standing open.