Two definitions of singular cohomology I am reading Hatcher's Algebraic Topology and Milnor's Characteristic Classes. In these two books, the definition of singular cohomology is little bit different, as follows: Fix a topological space $X$ and a (commutative) ring $R$ (with $1$). First, in Hatcher, we form a chain complex $\dots \to C_n(X) \xrightarrow{\partial} C_{n-1}(X)\to \cdots$ where $C_n(X)$ is the free abelian group with one generator for each singular simplex $\sigma\colon\Delta^n\to X$. Then we take $\text{Hom}(-,R)$ of this complex to obtain a cochain complex $\cdots \to \text{Hom}(C_{n-1}(X),R) \to \text{Hom}(C_n(X),R)\to \cdots$ . Hatcher defines the $n$-th cohomology group $H^n(X;R)$ from this complex.
On the other hand, in Milnor we form a chain complex $\dots \to C_n(X;R) \xrightarrow{\partial} C_{n-1}(X;R)\to \cdots$ where $C_n(X;R)$ is the free $R$-module with one generator for each singular simplex $\sigma\colon\Delta^n\to X$. Then we take $\text{Hom}_R(-,R)$ of this complex to obtain a cochain complex $\cdots \to \text{Hom}_R(C_{n-1}(X;R),R) \to \text{Hom}_R(C_n(X;R),R)\to \cdots$ . Milnor defines the $n$-th cohomology group $H^n(X;R)$ from this complex.
Is there no difference between these two definitions?
 A: I will elaborate a bit on Angina Seng's comment that there is a natural isomorphism $F\colon\text{Hom}_\mathbb{Z}(C_n(-), R) \to \text{Hom}_R(C_n(-;R), R)$ as functors to the category of $R$-modules. In essence this is due to the fact that the functors $\text{Hom}_\mathbb{Z}(-,R)$ and $\text{Hom}_R(-\otimes R, R)$ on the category of abelian groups are isomorphic.
First recall that $C_n(X;R) = C_n(X)\otimes_\mathbb{Z} R$. Given a group homomorphism $\varphi\colon C_n(X)\to R$, define the $R$-linear homomorphism $F(\varphi)\colon C_n(X;R)\to R$ by $F(\varphi)(g\otimes r) = \varphi(g)\cdot r$.
As an exercise you should verify for yourself that $F$ is a natural transformation of functors $\mathbf{Top}\to R\mathbf{-mod}$ (along the way you will need to verify that $F(\varphi)$ is well-defined wrt to the tensor product relation), and that for any given space $F$ is injective and surjective. Hints: for injectivity, once you establish that $F$ is an $R$-linear homomorphism for each space you only need to show $F(\varphi) = 0$ implies $\varphi = 0$, and for surjectivity if you are given an $R$-linear $\varphi'\colon C_n(X;R) \to R$, consider $\varphi'$ restricted to the subgroup $\{g\otimes 1\}$.
