All the shortest paths from top to top s at the top t: for graphs with positive weights

The code to search for all the shortest paths from top to bottom is provided in[BreadthFirstScan][1]. The relevant parts are under Applications, then Applications the shortest path, and in this section: the second example (for unweighted graphs). Is there online information for which an adaptation is given for graphics with edge weights (positive weights)? The code for the unweighted case is copied below.

Find all the shortest paths in an unweighted chart, with an example of a grid chart:

s = 1; t = 3 * 4;
g = GridGraph[{3, 4}, VertexSize -> {s -> Medium, t -> Medium}]

découvrirFun[u_, v_, d_] : = Si[u != v,  PropertyValue[{g, u},      
"ShortestPaths"] = Table[Append[p, u], {p, PropertyValue[{g, v}, 
"ShortestPaths"]}]; Value of the property[{g, u}, "Distance"] = d]revisiterFun[u_, v_] : = Si[Valeurdelapropriété[PropertyValue[Valeurdelapropriété[PropertyValue[{g, u}, "Distance"] ==
Value of the property[{g, v}, "Distance"] + 1, PropertyValue[{g, u}, 
"ShortestPaths"] = Join[PropertyValue[{g, u}, "ShortestPaths"],
Table[Append[p, u], {p, PropertyValue[{g, v}, "ShortestPaths"]}]]]Value of the property[{g, s}, "ShortestPaths"] = {{s}};
Value of the property[{g, s}, "Distance"] = 0;

LargeurAvantScan[g,  s, {"DiscoverVertex" -> discoverFun, 
"VisitedVertex" -> revisitFun,  "UnvisitedVertex" -> revisitFun}];

Table[HighlightGraph[g, p], {p, PathGraph / @ PropertyValue[{g, t},