J.S. Milne, in his electronic book : www.jmilne.org/math/CourseNotes/AGS.pdf , page : $ 65 $, says :
A subfunctor $ Z $ of a functor $ Y $ from $ \mathrm{Alg}_k $ to $ \mathrm{Set} $ is said to be closed if, for every $ k $ - algebra $ A $ and natural transformation $ h^A \to Y $, the fibred product $ Z \times_Y h^A $ is represented by a quotient of $ A $. The Yoneda lemma identifies a natural transformation $ h^A \to Y $ with an element of $ \alpha $ of $ Y(A) $, and, for all $ k $ - algebra $ R $, $ \Big( Z \times_Y h^A \Big) (R) = \{ \ \varphi \ : \ A \to R \ | \ \varphi ( \alpha ) \in Z(A) \ \} $.
Could you explain to me, how, by definition, is defined the functor : $ \Big( Z \times_Y h^A \Big) (R) $, which leads to his final description as : $ \Big( Z \times_Y h^A \Big) (R) = \{ \ \varphi \ : \ A \to R \ | \ \varphi ( \alpha ) \in Z(A) \ \} $ ?
Thus, $ Z $ is closed in $ Y $ if and only if, for every $k$ - algebra and $ \alpha \in Y(A) $, the functor of $k$ - algebras $ R \to \{ \ \varphi \ : \ A \to R \ | \ \varphi ( \alpha ) \in Z(A) \ \} $ is represented by a quotient of $ A $, i.e : there exists an ideal $ \mathfrak{a} \subset A $ such that, for all homomorphism $ \varphi \ : \ A \to R \ $ : $ \ Y ( \varphi ) ( \alpha ) \in Z(R) \ \ \Longleftrightarrow \ \ \varphi ( \mathfrak{a} ) = 0 $.
Could you explain to me, what does mean $ Y( \varphi ) ( \alpha ) $ in the last paragraph above, and why does it mean that the fact : "there exists an ideal $ \mathfrak{a} \subset A $ such that, for all homomorphism $ \varphi \ : \ A \to R \ $ : $ \ Y ( \varphi ) ( \alpha ) \in Z(R) \ \ \Longleftrightarrow \ \ \varphi ( \mathfrak{a} ) = 0 $" means that : $ R \to \{ \ \varphi \ : \ A \to R \ | \ \varphi ( \alpha ) \in Z(A) \ \} $ is represented by a quotient of $ A $ ?
Thanks in advance for your help.