Cartesian products in Set are categorical products

up:: Product (Category)

Given two sets $X_1,X_2$, denote their cartesian product as $X_1\times X_2$, and let

$${\pi }_i:X_1\times X_2\to X_i$$

be the respective projection maps.

Let $Y$ be another set.

Every function to the cartesian product is completely determined by the projection maps

Let $h:Y\to X_1\times X_2$ be any map.

Then it is uniquely determined by its values under composition with the projection maps, since

$$\forall y\in Y:({\pi }_i\circ h)(y)=\widetilde{x_i}(y)\in X_i$$

Functions to coordinate sets uniquely induce factor map to cartesian product

Conversely, let $f_i:Y\to X_i$ functions from $Y$ to each “coordinate” set $X_i$. Then there is a unique map $h:Y\to X_1\times X_2$ such that

$${\pi }_i\circ h=f_i:Y\to X_i$$

In fact, it is of the form

$$h(y)=(f_1(y),f_2(y))\in X_1\times X_2$$

which is uniquely determined by $f_i$ and which satisfies the condition above.

Conclusion

Note that to every function

$$h:Y\to X_1\times X_2$$

there is a unique function

$$\begin{array}{rl}{f}_1\times f_2:& Y\times Y\to X_1\times X_2\\ & (y,y)\mapsto \left(f_1(y),f_2(y)\right)\end{array}$$

where $f_i={\pi }_i\circ h:Y\to X_i$ are uniquely determined by $h$.

Thus, there is an isomorphism between the Hom-Sets $Hom(Y,X_1\times X_2)$ and $Hom(Y\times Y,X_1\times X_2)$.

---

References

• MAC LANE, Saunders, Categories for the Working Mathematician, New York, NY: Springer New York, 1978.