Chain homotopy between mapping cones

I am reading section 1.5 of Weibel's book on homological algebra. I understand the intuition of the mapping cone of a chain complex if we are talking about simplicial homology but I would like to know how broadly the argument generalizes.

Let $$f: X\to Y$$ ne a continuous map of topological spaces, and $$C_{\bullet}(X),C_{\bullet}(Y)$$ are the associated singular chain complexes, with $$f_\sharp :C_{\bullet}(X)\to C_{\bullet}(Y)$$ the induced morphism of chain complexes.

Let $$\operatorname{cone}{f}$$ be the mapping cone (as a topological space) formed by gluing the cone $$C(X)$$ to $$Y$$ along the relation $$(x,1)\sim 0$$ if $$f(x)=y$$.

Let $$\operatorname{cone}{f_\sharp}$$ be the chain complex mapping cone from homological algebra, as constructed in i.e. Weibel.

Are $$C_\bullet(\operatorname{cone} f)$$ and $$\operatorname{cone}(f_{\sharp})$$ chain homotopy equivalent? How can I prove this?

It is clear that the map $$X\to Y\to \operatorname{cone}f$$ is nullhomotopic. I have figured out how to construct a map from $$C_n(X)$$ to $$C_{n+1}(\operatorname{cone}f)$$ which implements this chain nullhomotopy. I am not sure how this helps me.