Would you share your system prompt for that grammatical checker?

There is no single prompt.

The languages I am learning have verb conjugations and noun declensions. So I write a prompt asking the LLM to break the given paragraphs down sentence-by-sentence by giving me the general sentence level English translation plus word-by-word grammar and (contextual) meaning.

For the grammar, I ask for the verbal root/noun stem, the case/person/number, any information on indeclinables, the affix categories etc.