 Cartesian closed category of rectangular bands

The category of rectangular bands is Cartesian closed. For nonempty products $A=B\times C$, $D=E\times F$, the exponential is $E^B\times F^C$. Also $D^0=1$ and $0^A=0$ for nonempty $A$. It has no subobject classifier: the classifier would have two elements, hence would be left-zero or right-zero, while its characteristic maps on a two-by-two product could not single out a one-point subalgebra.