How to find the smallest of the maximum edges of all paths between two nodes in a graph
In biref, you can just replace addition function with max function everywhere in the typical shortest path algorithm and it will work because they 'behave' the same way and that the algorithm only ever require the weight (together with the aggregation function) to have that 'behaviour'.

More specifically about that required 'behaviour' of weight.
The algorithm only require weight to be any type $\mathbb{W}$ with 'ordered monoid' structure, i.e.,
comes associated with an addition function
$$+\colon \mathbb{W} \times \mathbb{W} \rightarrow \mathbb{W}$$
and with a unique value
$$0 \in \mathbb{W}$$
such that it is associative and unital
$$(a+b)+c = a+(b+c)$$
$$0+a = a+0 = a$$
and finally with a binary (partial or total) ordering relation on $\mathbb{W}$ satisfying
$$x ≤ y \implies z+x ≤ z+y \land x+z ≤ y+z$$
You should check that the correctness proof of the shortest path algorithm only ever utilise these properties.

For fun, you could weight the edges of graph by string, add them using concatenation, have empty string be unit of concatenation, and ordered them lexicographically. Now you can find which path between 2 given points spells out the earliest word in dictionary using shortest path algorithm!