Product

• $A\times B$ with projections:

$${\pi }_1:A\times B\to A,{\pi }_2:A\times B\to B$$

• Universal property: For any $X$ with $f:X\to A,g:X\to B$,

$$\exists !⟨f,g⟩:X\to A\times B$$

such that:

$${\pi }_1\circ ⟨f,g⟩=f,{\pi }_2\circ ⟨f,g⟩=g$$