Building maps of singular chain complexes: is the identity singular complex $id:\Delta_q\rightarrow\Delta_q$ a kind of 'generator'?

Question

I noticed that in the proofs of two fundamental results of the theory of singular homology, such as the Homotopy Invariance Theorem and the Excision Theorem, we build maps between singular chain groups of some topological space $X$, $s_q:S_q(X)\rightarrow\cdot$, in the following fashion:

1. We consider the special case $X=\Delta_q$ and the identity singular complex $id:\Delta_q\rightarrow\Delta_q$ and define our map $s_q$ on it. So we get an element $s_q(id)$, in some other group of some other singular chain complex depending on the situation.
2. We extend the assignment to any $\sigma\in S_q(X)$ by naturality and linearity, thanks to the fact that $\sigma:\Delta_q\rightarrow X$ induces a morphism of chain complexes.

Is there something more general here? In which sense $\Delta_q$ ia a special object? Can we formalize it, and make it general using categories?

Thanks in advance