I want to solve an exercise about adjoints (but I am not sure about the steps).
Let $U:\textbf{CL}\rightarrow\textbf{Posets}$ be the forgetful functor. Here we have a functor from the complete lattices with joinpreserving maps to posets. Let $F$ be the functor the other way around defined by: $$F(X,\leq)=\{U\subseteq X:\forall x, y\in X: y\leq x\in U \Rightarrow y\in U\}$$
My question is: Why is the image under $F$ already an complete lattice? Why is F the left adjoint of U? Someone with hints of solutions?
Thanks :-)