Homotopy of continuous maps induces chain homotopy of corresponding chain maps

Statement
Suppose $$X$$ and $$Y$$ are topological spaces. Suppose $$f$$ and $$g$$ are continuous maps from $$X$$ to $$Y$$. We thus get chain maps $$C(f):C(X) \to C(Y)$$ and $$C(g):C(X) \to C(Y)$$.

Suppose $$H$$ is a homotopy from $$f$$ to $$g$$. Then, $$H$$ induces a chain homotopy between the chain maps $$C(f)$$ and $$C(g)$$.