One of the characterizations of $\kappa$ being a weakly compact cardinal is being inaccessible, and for every $\kappa$-model $M$, there is a [$\kappa$-model] $N$ and an elementary embedding $j\colon M\to N$ with critical point $\kappa$.

I am looking for a reference, or at least a proof, that this is equivalent to another property of weakly compact cardinals (which is not an embedding characterizations), for example the tree property or indescribability, or even the extension property.

However all the papers and books refer to one of two places:

1. Kanamori,
2. Cummings' Handbook article (section 16).

A diligent search through Kanamori produced nothing (the second edition, anyway). In Cummings' paper he refers to Kanamori, and only proves an implication from the embedding property to the Hauser embedding property (that we can also assume $j,M\in N$).

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> Does anyone knows where to find a proof for any equivalence of weak compactness? (Or if it is really folklore, *what* is the proof?)
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It should be easy to prove some characterizations *from* the embedding property, but I do not quite see how to prove the embedding property from something like compactness, the tree property, indescribability, or end-extensions.

(I should also say that I scoured other standard books like Jech and the Handbook, as well newer books like Schindler's Set Theory. To non-avail.)