Isomorphism of chain complexes In my notes it says $C^{sing}_n(\sqcup_{i\in I} X_i;R) \cong {\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$, where $C^{sing}_n$ denotes the n-th singular chain complex and $R$ is a ring, $S_n(X)$ is the set of all continuous maps $\sigma: {\Delta}_n \to X$ and ${\Delta}_n$ is the standard n-simplex. It is quite obvious that $S_n(\sqcup_{i\in I} X_i) = \sqcup_{i\in I}S_n(X_i)$, since the maps are continuous.
But if I take an element $ c \in C^{sing}_n(\sqcup_{i\in I} X_i)$, then I can write $c = {\sum}_{\sigma \in S_n(\sqcup_{i\in I} X_i)} {\lambda}_{\sigma} \sigma , {\lambda}_{\sigma} \in R$ and then $c = {\sum}_{\sigma \in S_n(\sqcup_{i\in I} X_i)} {\lambda}_{\sigma} \sigma = {\sum}_{\sigma \in \sqcup_{i\in I}S_n(X_i)} {\lambda}_{\sigma} \sigma = {\sum}_{i \in I}{\sum}_{\sigma \in S_n(X_i)} {\lambda}_{\sigma} \sigma \in {\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$. So this would suggest $C^{sing}_n(\sqcup_{i\in I} X_i;R) = {\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$. 
Where am I wrong? What is the explicit isomorphism (and why is it an isomorphism)?
 A: Whether these chain complexes are literally equal depends on the precise set-theoretic definitions you have chosen for all the notation involved.  For instance, a common definition of ${\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$ is the set of all functions $f$ with domain $I$ such that $f(i)\in C^{sing}_n(X_i;R)$ for each $i\in I$ and $f(i)$ is the zero element of $C^{sing}_n(X_i;R)$ for all but finitely many values of $i$.  Such a function is not an element of the set $C^{sing}_n(\sqcup_{i\in I} X_i;R)$.  Also, $S_n(\sqcup_{i\in I} X_i) = \sqcup_{i\in I}S_n(X_i)$ is not true for the usual definition of $\sqcup$ (which says something like $\sqcup_{i\in I} X_i=\bigcup_{i\in I} \{i\}\times X_i$).
In any case, worrying about these exact set-theoretic definitions is missing the point.  We don't actually care exactly what set the notation ${\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$ refers to, we only care about certain properties it has which characterize it up to canonical isomorphism.  The correspondence between elements of ${\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$ and elements $C^{sing}_n(\sqcup_{i\in I} X_i;R)$ which you have written down is an isomorphism, and that is what matters.  Maybe for some choices of the set-theoretic definitions involved your correspondence is actually literally the identity, but that doesn't really make a difference for anything you might want to do with it.
A: First, there's a mistake in in one of your formulas: $S_n(\amalg_{i\in I} X_i)\cong\bigoplus_{i\in I} S_n(X_i)$ and not disjoint union. Also, the two chain complexes you are interested in can't be exactly equal since the left hand side consists of linear combinations of chains, whereas the right hand side consists of $I$-tuplets of linear combinations of chains. The isomorphism between them is morally close to the identity though.
