Universal Property of Product of Topological Spaces

Theorem
Let $Y$ be a topological space.

Let $\left \langle {X_i}\right \rangle_{i \mathop \in I}$ be a family of topological spaces.

Let $X = \displaystyle \prod_{i \mathop \in I} X_i$ be the product space.

Let $\pi_i : X \to X_i $ denote the natural projections.

Let $\left \langle {f_i}\right \rangle_{i \mathop \in I}$ be a family of continuous mappings $f_i: Y \to X_i$.

Then their exists a unique continuous mapping:
 * $f : Y \to X$

which verifies $f_i = \pi_i\circ f$ for all $i\in I$.