Here is a little diagram inspired from MIT's 18.650 class which I find quite useful as it helps visualizing the relationships between these functions. I have used the same notation as in @momo's post:

• $\gamma(\theta)$$\gamma'(\theta)$ is the cumulant moment generating function
• $g(\mu)$ is the link function

So the link function $g$ relates the linear predictor to the mean and is required to be monotone increasing, continuously differentiable and invertible.

The diagram allows to easily go from one direction to the other, for example:

$$ \eta = g \left( \gamma(\theta)\right)$$$$ \eta = g \left( \gamma'(\theta)\right)$$ $$ \theta = \gamma'^{-1}\left( g^{-1}(\eta)\right)$$

Canonical link function

Another way of seing what Momo has described rigorously is that when $g$ is the canonical link function, then the function composition $$\gamma^{-1} \circ g^{-1}= \left( g \circ \gamma' \right)^{-1} = I$$$$\gamma'^{-1} \circ g^{-1}= \left( g \circ \gamma' \right)^{-1} = I$$ is the identity and so we get $$\theta = \eta $$