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Composition of Functions - in the context of morphisms in algebraic geometry

I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets.

On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows:

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Definition. A map is called a morphism (or polynomial map or regular map) of algebraic sets if

there are polynomials such that

for all

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D&F then go on to define a map between the quotient rings k[W] and k[V] as follows: (see attachment page 662)

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Suppose F is a polynomial in .

Then is a polynomial in

since are polynomials in .

... ... etc etc

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I am concerned that I do not fully understand exactly how/why .

I may be obsessively over-thinking the validity of this matter (that may be just a notational matter) ... but anyway my understanding is as follows:

so then we have that ...

.

**Can someone please confirm that the above reasoning and text is logically and notationally correct?**

Peter