Cartesian closedness of the category of real-valued sets, II

Let $[0,1{]}_{\ast }$ be the unit interval $[0,1]$ equipped with a left-continuous t-norm $\ast$. We prove that $[0,1{]}_{\ast }\text{-}\mathbf{S}\mathbf{e}\mathbf{t}$ is cartesian closed if and only if $\ast$ is the minimum t-norm. This extends the classification for continuous t-norms established in the first part of this work to the whole left-continuous setting.