I'm trying to put together the relationships between homotopy in $\mathsf{Top}$, chain homotopy, and homotopy in $\mathsf{Ch}_\bullet(R\mathsf{Mod})$. I more-or-less understand the connection between the former two and between the latter two, but I don't understand whether a homotopy in $\mathsf{Top}$ naturally induces a homotopy in $\mathsf{Ch}_\bullet(R\mathsf{Mod})$.
I can use the explicit form of the chain homotopy in Hatcher's proof of the homotopy invariance of singular homology and obtain from it a homotopy in $\mathsf{Ch}_\bullet(R\mathsf{Mod})$. This $$\text{homotopy in }\mathsf{Top} \rightarrow \text{chain homotopy} \rightarrow \text{homotopy in }\mathsf{Ch}_\bullet$$ process, while getting me what I want, leaves me hoping for a better alternative - the homotopy I get seems contrived because of all the games with simplices.
Is there any simpler homotopy on $\mathsf{Ch}_\bullet$ induced by a homotopy in $\mathsf{Top}$? Perhaps it has a simpler form?
Edit: I should clarify I'm looking for an induced homotopy in the sense of abstract homotopy theory, i.e in the sense of cylinder objects. I want an simple, intuitive, abstract homotopy $\phi:f_\sharp \simeq g_\sharp$ induced from a homotopy in $\mathsf{Top}$ $h:f\simeq g$.
A homotopy in spaces is a map $I \times X \to Y$. The Eilenberg-Zilber map can be used to induce from here a map $C_{\bullet}(I) \otimes C_{\bullet}(X) \to C_{\bullet}(Y)$. And with the right model of chains on the interval, this is the same thing as a chain homotopy.
There are ways to describe this construction without using chains or simplices, e.g. using the language of tensor (smash) products of spectra, but at some point you still need a theorem telling you how to actually compute stuff.