I think it depends how broadly or narrowly you conceptualise computer science. I doubt category theory is much use for developing algorithms, but it's been an important influence on type systems for programming languages, especially modern functional languages but also more mainstream things like generic types in Java.
Taking your question literally (“category theory being applied to solving a CS problem”), I'd say the full abstraction problem for PCF is a good example of a problem in computer science whose solution leant heavily on category theory.
Taking your question literally (“category theory being applied to solving a CS problem”), I'd say the full abstraction problem for PCF is a good example of a problem in computer science whose solution leant heavily on category theory.