What does this mean in the case of the usual adjunction between $ \mathbf{Grp} $ and $ \mathbf{Set} $ ? Given sets $ A $ and $ B $ , a partial function from $ A $ to $ B $ is a pair $ (S, f) $ consisting of a subset $ S \subseteq A $ and a function $ S \to B $ . (Think of it as like a function from $ A $ to $ B $ , but undefined at certain elements of $ A $ .)