No, this isn't right. You need to move the in the denominator upstairs (if you are thinking of this as an exponential dispersion family, you can't have the dispersion parameter be negative). The canonical parameter for this should be . The canonical link turns out to be the same - except as a function of the mean instead of the canonical parameter. This isn't true in general, of course (unless your base parameterization is always in terms of the mean, I suppose).
The exponential dispersion form of the density should be: .
It's also worth mentioning that people apparently give the canonical link of the exponential as instead of because these two are equivalent in terms of actually fitting the model and the negative sign looks ugly. But the canonical parameter DOES have the negative in it.