Some authors introduce singular homology as a family of homology functors indexed by $n\in\mathbb Z$:
$$H_n\colon \textbf{hTop} \to \textbf{Ab}$$
However, other authors define one functor
$$H\colon \textbf{hTop} \to \textbf{GradedAb}$$
Clearly both points in some sense "agree" on the resulting objects – a sequence $H_n(X)_{n\in \mathbb Z}$ is "essentially the same" as $\bigoplus_n H_n(X)$.
However, there is a slight difference in morphisms. A continuous map (or rather its equivalence class) $f\colon X\to Y$ induces a family of maps $H_n(f)\colon H_n(X)\to H_n(Y)$.
These can be "glued" into a morphism
$$\prod_n H_n(X) \to \bigoplus_m H_m(Y) = H_m(Y)$$
however usually $\prod_n H_n(X)$ is bigger than $H(X) = \bigoplus H_n(X)$ and such "glued" morphism does not need to be a graded morphism.
For spaces homotopy equivalent to finite CW-complexes almost all homology groups vanish and both point of views seem to be equivalent.
Which is the preferred notion of singular homology? Do they agree for all spaces, not only CW-complexes?
I prefer the first definition – not only it obviously works for all topological spaces, but also it homology can be defined in the setting of homological algebra using the language of derived functors. (And I'm not aware of any way of glueing resulting homologies in an arbitrary abelian category – for these only finite sums are guaranteed to exist).
Best Answer
A family of maps $H_n(f):H_n(X) \rightarrow H_n(Y)$ induce a map $\bigoplus_i H_i(X) \rightarrow \bigoplus_j H_j(Y)$. I don't know why you're introducing $\Pi_i H_i(X)$.
There is also no canonical map $\Pi_i H_i(X) \rightarrow \bigoplus_j H_j(Y)$, the canonical map would be $\bigoplus_i H_i(X) \rightarrow \Pi_j H_j(Y)$.