This question refers to J. Lurie's notes on Chromatic Homotopy Theory, Lecture 4. http://www.math.harvard.edu/~lurie/252xnotes/Lecture4.pdf
It is mentioned there that if $E$ is a multiplicative cohomology theory then $E^*(*)\cong \pi_0(E)$ is equipped wit a "canonical unit element" $\bar{t}$.
So the question is - how am I supposed to see this canonical unit element? I know that it should come from the unit map of spectra $\mathbb{S}\to E$, but I cannot see how I got a canonical unit element out of it.
An element of $\pi_0(E)$ is, by definition, a map from $\mathbb{S}\to E$. If you have a unit map, then take the corresponding element in $\pi_0$.