The catalogue is not the shelf
How Does a Library Find One Book?
A library cannot arrange its shelves by subject and by author at once, so it keeps a second thing that points.
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Step 1 of 5
Shelves can only be in one order.
A card is cheap to move. That is the entire reason catalogues exist. Once the books stand in subject order they are not in author order, and changing that means carrying every book. A shelf can answer exactly one kind of question quickly.
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Step 2 of 5
So the library keeps a list of pointers instead.
Every book gets a short entry: title, author, subject, and a number saying where the book stands. Entries are small enough to keep in several orders at the same time.
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Step 3 of 5
The pointer is what does the finding.
You search the catalogue, not the building. It hands you a number, and the number takes you to one shelf out of thousands without walking past any of the others.
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Step 4 of 5
One book can sit in many lists at once.
One copy on the shelf, three ways in. A book on the history of bridges belongs under history, under engineering and under its author. It cannot stand in three places, but its entry can appear in three lists.
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Step 5 of 5
Every search box you have used works this way.
A search engine does not read the web when you ask it something. It has already built a list of pointers from words to pages, and it answers out of that list.
The short version
A library finds one book by never searching the shelves: it keeps a small separate list of pointers, which can be held in several orders at once, and the pointer sends you straight to one place.
Try it yourself
Find the number on the spine of any library book. That number is the pointer, and it is the only part of the catalogue that has to be right for the book to be findable.