It is a well-known fact that a nondeterministic automaton can be transformed into an equivalent deterministic automaton via the powerset construction. From a categorical perspective this construction is the right adjoint to the inclusion functor from the category of deterministic automata to the category of nondeterministic automata. This is in fact an adjunction between two categories of coalgebras: deterministic automata are coalgebras over $\mathbf{S}\mathbf{e}\mathbf{t}$ and nondeterministic automata are coalgebras over $\mathbf{R}\mathbf{e}\mathbf{l}$. We will argue that this adjunction between coalgebras originates from a canonical adjunction between $\mathbf{S}\mathbf{e}\mathbf{t}$ and $\mathbf{R}\mathbf{e}\mathbf{l}$. In this paper we describe how, in a quite generic setting, an adjunction can be lifted to coalgebras, and we compare some sufficient conditions. Then we illustrate this technique in length: we recover several constructions on automata as liftings of basic adjunctions including determinization of nondeterministic and join automata, codeterminization, and the dualization of linear weighted automata. Finally, we show how to use the lifted adjunction to check behavioral equivalence.