I'm looking for references on the generalized cartesian product of graphs. More specifically, given a family $\{G_i\}_{i\in I}$ of (multidi)graphs with vertex sets $\{V_i\}_{i\in I}$ , one can define a graph $G=\prod_{i\in I} G_i$ with vertex set $V=\prod_{i\in I} V_i$ and the edges as following:
For any $i\in I$, vertices $v,w\in V_i$ and each edge $e$ from $v$ to $w$ there is a distinct edge $e'$ from $v'$ to $w'$ for each $v'\in \pi_i^{-1}(v)$ and $w'\in\pi_i^{-1}(w)$. Here "distinct" means that multiple edges between two vertices in a factor are accounted for with multiple edges between the respective vertices in the product accordingly.
Most literature only covers a finite amount of factors, usually of simple graphs, and the only generalized version I could find could be backtracked to Sabidussi, who only covers simple factors. I couldn't find references for the generalized form as above. Help would be appreciated.