I read* that the formula $$\diamond \varphi\rightarrow\square\diamond\varphi$$is valid in a structure $(W,R)$, intended as in Kripke semantics, -i.e. that it is true for any interpretation $I$ and in any world $u\in W$ of a model $(W,R,I)$- if and only if relation $R$ is Euclidean, i.e. if and only if relation $R$ is such that, if $uRw$ and $uRv$, then $vRw$ and $wRv$.
I would be interested in proving it to myself, but, although the converse is quite straightforward even for me, I cannot find a way to show that if $\diamond \varphi\rightarrow\square\diamond\varphi$ is valid then $R$ is Euclidean. I have tried to define a particular interpretation $I$ such that the Euclidean character of $R$ is shown, but I cannot find one. I have tried, for example: once a world $u\in W$ has been fixed, defining $I(P,w)=1$, i.e. $P$ true in world $w$, if and only if $w$ is accessible from $u$ ($ uRw$). But my trials have been fruitless until now. Could anybody explain how to prove that $\diamond \varphi\rightarrow\square\diamond\varphi$ is valid only if $R$ is Euclidean? Thank you very much!
*D. Palladino, C. Palladino, Logiche non classiche.