I am reading the paper On PL De Rham theory and rational homotopy type by Bousfield and Gugenheim. It was published in 1976, so some notation is kind of different to what I'm used. In particular, I'm unsure about the use of $\{,\}$ at page 8.
Let $\mathcal{S}$ be the category of simplicial sets, $\mathfrak{M}$ the category of $k$-modules. Given a contravariant functor $$K:\mathcal{S}\longrightarrow\mathfrak{M},$$ they define a new contravariant functor $\widehat{K}$ with the same domain and range by $$\widehat{K}(X) = \prod_{n\ge0}\prod_{x\in X_n}\{K(\Delta^n),x\}.$$ What are these brackets? Are they simply considering the $k$-module obtained by taking the product of one copy of $K(\Delta^n)$ for each element in $X_n$, that is $$\widehat{K}(X) = \prod_{n\ge0}K(\Delta^n)^{X_n},$$ or is it something completely different?
Edit: To add to the same question, they go on defining the action of $\widehat{K}$ on maps as follows. Let $f\in\mathcal{S}(X,Y)$, then for $m_y\in K(\Delta^n)$, $y\in Y_n$ we define $$\widehat{K}(f)\{m_y,y\} = \{m_{f(x)},x\}.$$ I can't make sense of this expression. I am trying to understand what the "obvious" action of $\widehat{K}$ on functions should be. My idea so far is that as $y\in Y_n$ can be seen as a morhism of simplicial sets $y:\Delta^n\to Y$, let $x\in X_n$ be such that $f(x) = y$. Then we can write down a commutative square $$\require{AMScd} \begin{CD} X @>f>> Y\\ @AxAA @AAyA\\ \Delta^n @>>\operatorname{id}> \Delta^n \end{CD}$$ and apply $K$ to it, remembering that it is contravariant: $$\require{AMScd} \begin{CD} K(X) @<K(f)<< Y\\ @VK(x)VV @VVK(y)V\\ K(\Delta^n) @<<\operatorname{id}< K(\Delta^n) \end{CD}$$ This makes me think that we should rather have something like $$\widehat{K}(f)\{m_y,y\} = \prod_{x\in f^{-1}(y)}\{m_y,x\},$$ where we see $m_y\in K(\Delta^n)$, which was associated to $y$, as an element of the copy of $K(\Delta^n)$ associated to $x$. Does this make sense?
Yes, that's right, $\hat{K}(X) = \prod_{n \ge 0} K(\Delta^n)^{X_n}$. The extra $x$ in the notation is just to know which factors comes from where. I think today people would rather write something like $\{ m_y \}_{y \in Y_n, \, n \ge 0}$, so that's what I'll do.
Now suppose that you have a simplicial map $f : X \to Y$. You want to define a map $\hat{K}(Y) \to \hat{K}(X)$. An element of $\hat{K}(Y)$ is a collection $\{ m_y \}_{y \in Y_n, \, n \ge 0}$, and $m_y \in K(\Delta^n)$ for $y \in Y_n$. To this you want to associate a collection $\{ m'_x \}_{x \in X_n, \, n \ge 0} \in \hat{K}(X)$ (where $m'_x \in K(\Delta_n)$). What they do is they simply set $m'_x = m_{f(x)}$.
(Reading this section, it seems they're implicitly taking the methods of acyclic models for granted, it might help you to read a bit about it before reading their work. I've written a blog post about it if you want, and there are references at the end.)