Group Direct Product is Product in Category of Groups

Theorem
Let $\mathbf{Grp}$ be the category of groups.

Let $G$ and $H$ be groups, and let $G \times H$ be their direct product.

Then $G \times H$ is a binary product of $G$ and $H$ in $\mathbf{Grp}$.

Proof
Let $F$ be a group.

By Direct Product of Group Homomorphisms is Homomorphism, given group homomorphisms:


 * $g: F \to G, h: F \to H$

their direct product $g \times h: F \to G \times H$ is a group homomorphism.

From Projections on Direct Product of Group Homomorphisms, the following diagram is commutative:


 * $\begin{xy}\xymatrix@L+3mu@R=3em{

& F \ar@/_/[dl]_*{g} \ar[d]^*{\quad g \times h} \ar@/^/[dr]^*{h} \\ G & G \times H \ar[l]^*{\mathrm{pr}_1} \ar[r]_*{\mathrm{pr}_2} & H }\end{xy}$

By Cartesian Product is Set Product, $g \times h$ is the only mapping $F \to G \times H$ that could fit into the diagram.

Since it is also a group homomorphism, we see that the UMP for the binary product is satisfied.

Thus the direct product $G \times H$ indeed is a categorical product for $G$ and $H$ in $\mathbf{Grp}$.