These are isomorphic homologies, and I found a sketch of a proof here on p. 53.
The proof is mostly exercises, done in steps first for finite complexes, then for infinite ones, using the fact that homology commutes with direct limits. The only experience I have with theories of homology is that I am currently reading Rotman, and doing most of the exercises. When I read the statement and proof of the Acyclic Carrier theorem, it occurred to me that there might be a much easier proof of the aforementioned isomorphism-so my proof is probably wrong. But I would like feedback on my idea:
Let $K$ be a simplicial complex, whose vertices have been partially ordered in some way. Now, $C_n(K)$ and $C_n^{or}(K)$ are the free abelian groups of oriented and ordered, resp., $n$-simplices. Write the generators $[v_1,\cdots, v_n]$ and $<v_1,\cdots,v_n>,\ $ resp., and define
$f:C_n(K)\to C_n^{or}(K):[v_1,\cdots, v_n]\mapsto <v_1,\cdots,v_n>$ ; $g:C_n^{or}(K)\to C_n(K):<v_1,\cdots,v_p>\mapsto \begin{cases} [v_1,\cdots, v_n]& v_i\neq v_j;\ 0\le i\neq j\le n \\ 0& \text{otherwise} \end{cases}$
Extend $f$ and $g$ by linearity. $f$ and $g$ are evidently augmentation-preserving chain maps. Furthermore,
$\tag 1 g\circ f=1_{C_n(K)}$
Now, define a function $E$ that maps the basis element $\gamma= <v_1,\cdots,v_n>$ of $C_n^{or}(K)$ to $E(\gamma)$, the subcomplex consisting of $\gamma$ and its faces. Then $E$ is a carrier function for $f\circ g$ and $1_{C^{or}(K)}$ so the Acyclic Carrier theorem gives
$\tag2 f\circ g\sim 1_{C^{or}(K)}$
Combining $(1)$ and $(2),$ we conclude that $f$ and $g$ are chain homotopy inverses, so that $f_*$ is an isomorphism.