We note that Bénabou writes “adjonction cartésienne” to refer to the product-exponential adjunction $\prn{-}\times X \dashv \prn{-}\Sup{X}$. Although this terminology is not current, we have retained it as “the cartesian adjunction”.

Another aspect of Bénabou’s terminology that we have retained is the use of “adjoint” to mean “adjoint transpose”; for instance, Bénabou speaks of the “cartesian adjoint” $T\times X\to Y$ of a morphism $T\to Y\Sup{X}$.