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# Question 2.7 - The product of categories

edited January 2020

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.