# Why does the canonical parameter give a link function? Why does this relate $E[Y]$ to $x^T \beta$?

If I have a pdf in the form $$f(y|\theta,\phi)=\text{exp}\bigg(\frac{y\theta-b(\theta)}{a(\phi)}+c(y,\phi)\bigg)$$, then $$\theta$$ is called the canonical parameter. I'm told we can get a link function $$g$$ from $$\theta$$ that links $$E[Y]$$ to $$x^T\beta$$, like so: $$g(E[Y])=x^T\beta$$.

E.g. For $$Y\sim Poisson(\mu)$$, $$P(Y=y)=\exp(-\mu+y\log \mu-\log y!)$$. From this, $$\theta=\log\mu$$. Hence, the link function is the log function: $$\log E[Y]=x^T\beta$$.

At the moment it seems like a total coincidence that the canonical parameter gives a function which just so happens to relate $$E[Y]$$ to a linear combination of some independent variables. Why does the canonical parameter have this property?