given a topological space $X$, $H_n(X,-)$ is a functor from the category of abelian groups to itself. i want to clarify the following :
1) given an homomorphism $f:G\rightarrow H$ of abelian groups what is explicitly the induced map $f_*:H_n(X,G)\rightarrow H_n(X,H)$
my guess: an element in $H_n(X,G)$ is a formal sum $\sum{g_i c_i}$ where $g_i\in G$ and $c_i$ a class of a cyle in $C_n(X,G)$ the free $\mathbb Z$-module on $n$-singular simplices. so $f_*(\sum{g_i c_i})=\sum{f(g_i) c_i}$
2)given an abelian group $A$, what is the canonical homomorphism $f:\mathbb Z \rightarrow A$ that is used to induce $f_*:H_n(X,\mathbb Z)\rightarrow H_n(X,A)$ and then induce $f_{**}:H_n(X,\mathbb Z)\otimes A\rightarrow H_n(X,A)$ that gives the short exact sequence in the universal coefficient theorem in homology:
$0 \rightarrow H_i(X, \mathbb{Z})\otimes A\rightarrow H_i(X,A)\rightarrow\mbox{Tor}(H_{i-1}(X, \mathbb{Z}),A)\rightarrow 0$