Suppose $K$ be a simplicial complex and denote the group of $p-\text{chains}$ from the set of oriented $p-\text{simplices}$ of $K$ to $\mathbb{Z}$ by $C_p(K)$, $p>0$. Let the set of oriented $p-\text{simplices}$ of $K$ be denoted by $S$.
If $\sigma$ is an oriented simplex, then the elementary chain $c$ corresponding to $\sigma$ is a function of the form $c:S\to \mathbb{Z}$ such that \begin{equation}c(\sigma)=0, c(-\sigma)=0 \text{ and } c(\tau)=0 \text{ for all other oriented simplices } \tau\end{equation}
We know that $C_p(K)$ is a free abelian group whose basis elements are elementary chains. We can denote the elementary chain corresponding to $\sigma$ by $c_\sigma$ and Munkres writes $c=\sum n_i\sigma_i$ by abuse of notation, using $\sigma_i$ to mean $c_{\sigma_i}$.
We wish to draw a homomorphism from $C_p(K)$ to $C_{p-1}(K)$. The author writes down the map $\partial_p$ where $\partial_p$ takes a p-simplex $\sigma$ (instead of a $p-\text{chain}$), written $[v_0,\dots,v_p]$ to a $p-1$-simplex $\displaystyle \sum_{i=0}^{p}(-1)^i[v_0,\dots,\hat{v_i},\dots,v_p]$ where a hat over vertex $v_i$ denotes that we consider the simplex generated by $v_0,\dots, v_p$ minus $v_i$.
Clearly, there is abuse of notation again. However, I am unable to pin down the map $\partial_p$ explicitly. There is, of course, a one-to-one correspondence of each $\sigma$ with $c_{\sigma}$. So, should $\partial_p(c_{[v_0,\dots,v_p]})$ equal $\displaystyle \sum_{i=0}^{p}(-1)^ic_{[v_0,\dots,\hat{v_i},\dots,v_p]}?$
thanks!