_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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