I stumbled upon this Lemma:

Let $X$ be a spectrum and $H_p(X;\Omega_q^{Spin})\Rightarrow MSpin_{p+q}(X)$ the Atiyah-Hirzebruch spectral:

The differential $d_2\colon H_p(X;\Omega_1^{Spin})\to H_{p-2}(X;\Omega_2^{Spin})$ is the dual of $Sq^2\colon H^{p-2}(X;\mathbb{Z}_2)\to H^p(X;\mathbb{Z}_2)$

The differential $d_2\colon H_p(X;\Omega_0^{Spin})\to H_{p-2}(X;\Omega_1^{Spin})$ is reduction mod $2$ (denoted with $r$) composed with the dual of $Sq^2$

And the proof is based on the following observation: $d_2$ is a stable homology operation and thus induced from elements in $$[H\mathbb{Z}/2, \Sigma^2 H\mathbb{Z}/2]\cong \mathbb{Z}/2\langle Sq^2\rangle$$ or $$[H\mathbb{Z}, \Sigma^2 H\mathbb{Z}/2]\cong \mathbb{Z}/2\langle Sq^2\circ r\rangle$$

where $HR$ is the Spectrum representing singular cohomology with coefficient $R$.

In this paper, def 3.1 suggest that an Homology Operation is a natural transformation between homology functors, but for example Switzer introduces the Homology Cooperations in a complete different way (they might be different but it seems that in some cases they re the dual of the cohomology operations). So my questions are:

**1)** First of all, what's the relationship between homology operations and the homology cooperations described in Switzer's book for example.

**2)** Why are we looking at the stable Cohomology operations and then dualise it?

**3)** later in the paper the following reasoning is done: the edge homomorphism for the AHSS for stable homotopy is a stable homology operation form stable homotopy to singular homology, i.e. an element of $[S^0,H\mathbb{Z}]\cong \mathbb{Z}\langle h \rangle$ where $h$ is the Hurewicz homomorphism. Therefore (after testing with certain space) we can conclude that the edge homomorphism is given by the Hurewicz map. Where can I see some reference for the fact that $[S^0,H\mathbb{Z}]\cong \mathbb{Z}\langle h \rangle$? why aren't we dualise anything here as done above? (I know that homology is not necessarily the dual of cohomology, but I want to understand why we dualise above and not here, since map of spectra indices stable Cohomology operations.