Question 2.7 - The product of categories

The product of categories.

Given two categories \(C\) and \(D\), we may construct a new category \(C \times C\) by taking pairs of objects and morphisms.. More precisely:

  • The objects of \(C \times D\) are pairs \((c,d)\) where \(c \in Ob\ C\) and \(d \in Ob\ D\).
  • The morphisms \((c_1, d_1) \rightarrow (c_2,d_2)\) are pairs \((f,g)\) where \(f: c_1 \rightarrow c_2\) in \(C\) and \(g: d_1 \rightarrow d_2\) in \(D\).
  • Composition is given pointwise: given \((f,g): (c_1,d_1) \rightarrow (c_2,d_2)\) and \((h,k): (c_2,d_2) \rightarrow (c_3,d_3)\), their composite is \((h \circ f, k \circ g): (c_1,d_1) \rightarrow (c_3,d_3)\).
  • Similarly, the identity morphisms are given by \((id_c,id_d): (c,d) \rightarrow (c,d)\).

Recall that the category Cat whose objects are categories and morphisms are functors. Show that \(C \times D\) is the product of \(C\) and \(D\) in Cat.

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