Corollary 3

Let X and Y be topological spaces and let $f: X\rightarrow Y$ be a continuous and relatively connected mapping. Let F be a set and let $g: X\rightarrow F$ be a locally constant mapping. There exists a unique locally constant mapping $h: Y\rightarrow F$ such that $g=h\circ f$.

The restriction of $g$ to every connected component of X is constant. By Corollary 1, there exists a mapping $h: Y\rightarrow F$, constant on each connected component of Y, such that $g=h\circ f$. The mapping $h$ is locally constant, since the connected components of Y are open. The uniqueness of such a mapping follows from the fact that $f(X)$ is dense in Y.