Background:
===========


$FGL(R)$ will be the category of formal group laws over the ring $R$. If $(E,x\_E)$ is an oriented spectrum (i.e. $x\_E \in \tilde{E}^2(\mathbb{P}^{\infty})$ is sent to the unit $1 \in \tilde{E}^2(\mathbb{P}^1)$ by the pullback of the inclusion $i: \mathbb{P}^1 \hookrightarrow \mathbb{P}^\infty$), then the pullback of the multiplication $\mu$ as H-space of $\mathbb{P}^\infty$ define an element in $FGL(E^\*(\*))$ in the following way:


$
\mu^\*:E^\*(\*)[[x\_E]] \cong E^\*(\mathbb{P}^\infty) \to E^\*(\mathbb{P}^\infty \times \mathbb{P}^\infty) \cong E^\*(\*)[[x\_1,x\_2]]
$


where $x\_i : = p\_i^\*(x\_E)$ and the isomorphism are given by the collapsing of the Atiyah Hirzebruch spectral sequence. The fgl associated to $(E,x\_E)$ is by definition $\mu^\*(x\_E) \in FGL(E^\*(\*))$. The Landweber exact theorem 'reverse' this contruction: let $f \in FGL(R)$ be a formal group law over $R$, then by certein hypothesis on $f$ the association


$ X \to MU^\bullet(X)\otimes\_{\pi\_\*(MU)} R$


is a cohomology theory on CW-complexes ($MU$ is the complex cobordism spectrum and the $\pi\_\*(MU)$-module structure on $R$ is given by the classification map $\pi\_\*(MU) \to R$, since by Quillen's theorem the fgl of the complex cobordism is the universal one). This correspondence given by the Landweber's theorem seems to be only objectwise.


Question:
---------


Given a morphism of formal group laws $\varphi$ over $R$, which is a power series in $R[[x]]$, is there a way to produce a natural transformation of the related cohomology theories induced by $\varphi$? Is there any reference in the literature where i could find an answer?



Morphism between formal groups and cohomology theories
Yes, if $E$ and $E'$ are Landweber exact, then ring maps $E\to E'$ biject with isomorphisms of the associated formal groups, suitably interpreted. One version of this (for the context where $E$ and $E'$ are $2$-periodic) is explained in Proposition 8.43 of [this Memoir](https://arxiv.org/abs/math/0011121). There is also a rather terse account in Proposition 11 of [this lecture of Lurie](http://www.math.harvard.edu/~lurie/252xnotes/Lecture18.pdf).


