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Exercise 86 - Chapter 3

Check that a product \( X \overset{p_X}{\leftarrow} X \times Y \overset{p_Y}{\rightarrow} Y \) is exactly the same as a terminal object in \( \textbf{Cone}(X, Y) \).

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  • 1.

    A terminal object in Cone\((X,Y)\) is a pair of morphisms \(X\xleftarrow[]{\pi_1}X\times Y\xrightarrow[]{\pi_2}Y\) such that for every pair of morphisms \(X\xleftarrow[]{f}C\xrightarrow[]{g}Y\), there is a unique morphism \((f,g):C\to X\times Y\) such that \(\pi_1\circ(f,g)=f\) and \(\pi_2\circ(f,g)=g\). This is exactly the same as the definition of the product of \(X\) and \(Y\) (cf. Definition 3.81).

    Comment Source:A terminal object in **Cone**\\((X,Y)\\) is a pair of morphisms \\(X\xleftarrow[]{\pi_1}X\times Y\xrightarrow[]{\pi_2}Y\\) such that for every pair of morphisms \\(X\xleftarrow[]{f}C\xrightarrow[]{g}Y\\), there is a unique morphism \\((f,g):C\to X\times Y\\) such that \\(\pi_1\circ(f,g)=f\\) and \\(\pi_2\circ(f,g)=g\\). This is exactly the same as the definition of the product of \\(X\\) and \\(Y\\) (cf. Definition 3.81).
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