Let $D(x) :=$ "$x$ is a dog", $C(x) :=$ "$x$ is a cat", and $F(x,y) :=$ "$x$ displeases $y$".
I proceeded as follows:
$\forall x(x \text{ is a cat and every dog } y \text{ displeases } x \rightarrow x \text{ displeases some dog } w)$.
At first, I was in doubt about the "every" in the antecedent. I thought it might meant "some", if that makes sense at all. I'm not a native english speaker, so I was afraid I may misinterpreted something. Nevertheless, I arrived at the following:
$\forall x [C(x) \land \forall y (D(y) \rightarrow F(y,x)) \rightarrow \exists w (D(w) \land F(x,w))]$,
so that the negation is
$\exists x [C(x) \land \forall y (D(y) \rightarrow F(y,x)) \land \forall w (D(w) \rightarrow \lnot F(x,w))]$.
Thoughts?