Prove that checking if context free grammar generates word with all symbols is NP-complete - unclear statement about encoing probem in exercise Prove that following problem is complete in NP in sense of Karp.
Given: context free grammar $G$
Check if $G$ generates word containing
all symbols from alphabet
Remark
Above problem should be encoded as langauge over finite alphabet,  although alphabet of $G$ can be arbitrarily big. You can assume that symbols of this alphabet are encoded as binary strings.

My main problem at this moment is that I don't understand Remark.  For me, it should be attached also alphabet of grammar.  Can you explain me in human language what this Remark try to say ?
To be more clear I can say that in other task with the same question, but with given regex and alphabet of regex I did deal with it. However, in mentioned task there was no this awkward Remark.
What the remark is trying to tell you, I think, is that the alphabet of each different grammar $G$ can grow and is not fixed.

For instance. say that you have a language $L$ that you want to reduce to your language, let´s call it $L'$, then, you are allowed to make the alphabet of the different context free grammars inside $L'$ grow as the instances inside $L$ grow. For a bigger instance a bigger alphabet