For small categories $\displaystyle A $, $\displaystyle B $, and $\displaystyle C $ establish a bijection $\displaystyle \bold{Cat}(A \times B, C) \simeq \bold{Cat}(A, C^{B}) $ and show it is natural in $\displaystyle A,B $ and $\displaystyle C $. Hence show that $\displaystyle - \times B: \bold{Cat} \to \bold{Cat} $ has a right adjoint.