I am reading Hatcher(alg.top).
in the chapter 3, section 3.1, the universal coefficient theorem, there it is being argued that,
In the original chain complex the homology groups are $\mathbb{Z}$'s in dimensions 0 and 3, together with a $\mathbb{Z}_2$ in dimension 1. The homology groups of the dual cochain complex, which are called cohomology groups to emphasize the dualization, are again $\mathbb{Z}$’s in dimensions 0 and 3, but the $\mathbb{Z}_2$ in the 1 dimensional homology of the original complex has shifted up a dimension to become a $\mathbb{Z}_2$ in 2 dimensional cohomology.
I don't quite understand where does this $\mathbb{Z}_2$ come from in the description. The confusion may be due to the fact I don't understand what the relation showed as a "vertical-equal" sign. If that means to "isomorphic to" or "Homology of" then I don't understand how $\mathbb{Z}_2$ comes to the description
Have a look at the top sequence. Kernel of the last $0$ map $C_1\to C_0$ is of course whole $\mathbb{Z}$. While the image of $x\mapsto 2x$ map is $2\mathbb{Z}$. The corresponding homology is the quotient of kernel by image, and gives us $\mathbb{Z}/2\mathbb{Z}$, also commonly referred to as $\mathbb{Z}_2$.
The vertical equality signs are indeed isomorphisms. But not of homology, but of (co)chain groups, the elements of those sequences. Well, for the top sequence these can be literal equalities. But for the bottom one these are isomorphisms, after applying $Hom$. The $x\mapsto 2x$ also gets transformed through $Hom$, but gives the same map (with reversed arrow) after applying the isomorphism. While zero morphisms are always mapped to zero morphisms through $Hom$.