In the book Étale Cohomology James Milne defines Henselization for a local Noetherian ring by the universal property:
Let $i: A \to A^\text{h}$ be a local homomorphism of local rings; $A^\text{h} $ is the Henselization of $A$ if it is Henselian and if any other local homomorphism from $A$ into a Henselian local ring factors uniquely through $i$.
It is clear that if $A^\text{h}$ exists then it it must be unique. Milne proves the existence by constructing the Henselization as a direct limit of étale neighborhoods of $A$.
However right after this in Exercise 4.9 he claims that it is also possible to construct the Henselization from above instead from below:
Let $\widetilde{A}$ be the intersection of all local Henselian subrings $H$ of $\hat{A}$, containing $A$, which have the property that $\hat{\mathfrak{m}} \cap H = \mathfrak{m}_H$. Show that $(\widetilde{A}, i)$, where $i: A \to \widetilde{A}$ is the inclusion map, is a Henselization of $A$.
Using the hint it becomes obvious that $\widetilde{A}$ is indeed Henselian. It is also clear that it satisfies the universal property for all rings inside $\hat{A}$. However I don't understand how to prove the universal property for all Henselian rings. Maybe someone here can explain me how this is done.
Thank you for advice in advance.