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# Exercise 86 - Chapter 3

edited June 2018

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 $$\textbf{Cone}(X, Y)$$.

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).