_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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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.

***

[Next](https://forum.azimuthproject.org/discussion/2417/question-2-8-bifunctors)

[Prev](https://forum.azimuthproject.org/discussion/2415/question-2-6-products-in-hask)

[All](https://forum.azimuthproject.org/categories/exercises-applied-category-theory-course)