The question is in the process of proving the statement in “Abstracte and Concrete Categories” book http://katmat.math.uni-bremen.de/acc/acc.pdf from the $\mathbf E\mathbf x. 5\mathbf E (a)$ on the page 78. The first and the second statements.
They are here. I linked the book to help others to use its definitions.
Show that no proper subconstruct of $\mathbf G \mathbf r\mathbf p$ is concretely reﬂective (or coreﬂective). Generalize this to all ﬁbre-discrete concrete categories.
As for me, the only way to find the proof is to come to contradiction, but I do not have any idea to find them. Of course, we need to use somehow the knowledge that this category is fibre-discrete but through the concrete reflector it remains to be fibre-discrete. Does someone know how to prove this statement?