After noticing an error in my math book, I rewritten its section “Funcoids and filters” to reflect that $latex (\mathsf{RLD})_\Gamma = (\mathsf{RLD})_{\mathrm{in}}$.
Previously I proved an example demonstrating that $latex (\mathsf{RLD})_\Gamma \ne (\mathsf{RLD})_{\mathrm{in}}$, but this example is believed by me to be wrong. The example was removed from the book.
Thus I removed all references to $latex (\mathsf{RLD})_\Gamma$ (as it is the same as $latex (\mathsf{RLD})_{\mathrm{in}}$) and reworked the chapter “Funcoids and filters” to reflect the change.
The book is available free of change at this Web page.
The story of the past:
$latex (\mathsf{RLD})_\Gamma$ was defined by the formula $latex (\mathsf{RLD})_\Gamma f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^\Gamma\, f$.
From the theorem in “The diagram” section (the theorem with a diagram) it trivially follows that $latex (\mathsf{RLD})_\Gamma f = (\mathsf{RLD})_{\mathrm{in}} f$. It follows trivially, but I have found this only today.
I’ve added to the post a note with my old definition.