Proposition 8.2

Proposition 8.2 (Universal property of the inverse limit). Let $Y$ be a set and suppose maps $f_n:Y\to X_n$ form a compatible cone: $$\rho_{n+1,n}\circ f_{n+1}=f_n \qquad\text{for every }n.$$ There is a unique map $$f:Y\longrightarrow\varprojlim_nX_n$$ such that $\pi_n\circ f=f_n$ for every coordinate projection $\pi_n$. If the spaces are topological and every $f_n$ is continuous, then $f$ is continuous.