I am currently writing my master's thesis about Whitehead's problem in homological algebra. My main source is Eklof's article https://www.jstor.org/stable/2318684. Eklof does not name the Diamond principle in the article (though I know Shelah uses it in his original proof), but instead uses the following axiom, which he lists as a consequence of V=L.
Let C be a set which is the union of a strictly increasing smooth chain of countable sets $\{C_{\upsilon}\ |\ \upsilon<\omega_{1}\}$ (smoothness means that $C_\lambda = \bigcup_{\upsilon<\lambda} C_\upsilon$ for $\lambda$ a limit ordinal), and let $E$ be a stationary subset of $\omega_{1}$. Then there is a sequence $\{S_{\upsilon}\ |\ \upsilon\in E\}$ such that $S_{\upsilon}\subseteq C_{\upsilon}$ for all $\upsilon\in E$ and such that for any subset $X$ of $C$, the set of $\upsilon\in E$ with $X\cap C_{\upsilon} = S_{\upsilon}$ is stationary in $\omega_{1}$.
On my first read of the article, I assumed this was either equivalent to or a simple consequence of the Diamond Principle. However, after actually trying to prove it, I'm not so sure anymore. I have not been able to construct a proof, and I'm not even sure this is actually the Diamond Principle. Are there any users out there who might shed some light on this? If this axiom is a consequence of $\diamondsuit$, how do I prove this? If not, how is it related to $\diamondsuit$?
To see that $\lozenge$ follows from this principle, simply set $C=\omega_1$ with $C_\nu=\nu$ and take $E=\rm Lim$, the club of limit ordinals. Then for every subset of $C$, that is $\omega_1$, the set of correct guesses is stationary.
In the opposite direction, it seems to me that Eklof defines $\lozenge_E$, which in general is stronger than $\lozenge$. Of course, under $V=L$ none of this matters, and ultimately, this is just a question of what is the most convenient tool for explaining Shelah's proof.