categorical product

{{context|category theory}} Given two objects ''A'' and ''B'', their '''product''' is an object <math>A \times B </math>, with projections <math>\pi_1: A \times B \rightarrow A</math> and <math> \pi_2: A \times B \rightarrow B</math>, which satisfies the following ''universal property'': for any object ''X'' with arrows <math>f: X\rightarrow A</math> and <math>g: X\rightarrow B</math>, then there can naturally be constructed an arrow <math> \langle f,g\rangle: X\rightarrow A \times B</math> such that <math>\pi_1 \circ \langle f,g\rangle = f</math> and <math>\pi_2 \circ \langle f,g\rangle = g</math>, and this arrow is unique (in satisfying these two equations).