If your father-in-law is not just posting but sharing his views in conversation, then yeah, of course your daughters are going to be aware of those views eventually. I don't know precisely what your father-in-law is saying or sharing, but "hate-filled" makes me imagine some pretty gross stuff.
I think it's very important for our kids to see us challenging those things, no matter who they're coming from. You and your husband can and should push back if your father-in-law says something you think is hateful or just out of line.
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