Implication Details

Claim: If a category is complete and is essentially small and is infinitary distributive and is thin, then it is cartesian closed.

Proof: This is an application of the adjoint functor theorem. Specifically, if PP is a complete lattice in which sup⁡iinf⁡(t,xi)=inf⁡(t,sup⁡iyi)\sup_i \inf(t,x_i) = \inf(t, \sup_i y_i) always holds, then the functor inf⁡(t,−)\inf(t,-) is a left adjoint because it preserves all suprema.