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

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.