They just show up everywhere, adjoints. Often you'll be able to then use their uniqueness and limit preservation properties. Typically CS style category theory is a little bereft of categories to make maximal use of their appearance, but they're there.
I gave a talk at LambdaConf this year mentioning that Free/Forgetful adjoints are a great way to understand free structures. There's also the fact that for all/exists arise as adjoints and you can use this to draw immediate conclusions like product preservation.
I gave a talk at LambdaConf this year mentioning that Free/Forgetful adjoints are a great way to understand free structures. There's also the fact that for all/exists arise as adjoints and you can use this to draw immediate conclusions like product preservation.