What does the phrase "be to Noun" mean?

Question

"Self-schemas are to an individual’s total self–concept as a hypothesis is to a theory, or a book is to a library." (source: Social Psychology by Steven Fein)

Subject is/are to Noun. I could not figure out the meaning of "be to noun" so I looked it up but could not find the meaning. What does the preposition "to" here mean? It cannot be "about or applied to".

In other words, analogy or another example to say the same thing etc, that I understood already. What I meant was I did not understand that "Self-schemas {are to} an individual’s total self–concept." Be to a verb is the idiom I understand but not this idiom in the bracket { }. (e.g. I understand that my hobbies are to read books, and to watch educational documentaries.) Could you help me clarify it?

Answer 1

Your question is worded in a confusing way, but I think what you're referring to is an analogy. (At least, the example you provided is an analogy.)

From Wikipedia:

An analogy can be stated using is to and as to represent the analogous relationship between two pairs of expressions, for example, "Smile is to mouth, as wink is to eye."

I think you changed "is" to "be" in your question, and when you were looking it up. You can't really conjugate "is" to "be" in the case of analogies. They are a particular idiomatic use of "is to" and "as to".

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Here's a list of simple analogies, if you want to see how they're used to relate different concepts: http://www.home-speech-home.com/word-analogies.html

Answer 2

What you've got there is something I would refer to as a grammar pattern. Its purpose is to draw a comparison between two pairs of things (one pair contains A and B and the other one contains D and C) to show how one thing (A) is important to another thing (B) in the first pair by contrasting that relationship with a similar one that exists between the two other things (D and C) in the second pair. This is how you could rewrite this grammar pattern in a more schematic form:

A is to B as D is to C

That to you were talking about basically means with respect to here. This is what the sentence is saying: A with respect to B is the same thing as D with respect to C. An example will help make this point clearer:

Water is to fish as land is to people.

What we're trying to say here is how important water is to fish by comparing that relationship to the idea of how land is important to human beings.