Given the exponential distribution: $\displaystyle f(x)= \theta^{-1}e^{-x/\theta$.

Converts it into exponential family form: $\displaystyle exp[\frac{(1/\theta)x-log(1/\theta)}{-1}]$

Isn't the canonical parameter and link function the same?

Which is $\displaystyle \frac{1}{\theta}$

Are there any differences between canonical parameter and canonical link function?