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 $P$ is a complete lattice in which $\sup_i \inf(t,x_i) = \inf(t, \sup_i y_i)$ always holds, then the functor $\inf(t,-)$ is a left adjoint because it preserves all suprema.

This implication has a dual.