What happens if I replace $<$ with $\le$ in Dijkstra's algorithm?
I would argue that this does not matter. Consider the case where there are multiple shortest paths from $s$ to $t$ in $G$. In the case of using $&lt;$, we define the shortest path to be the first shortest path we come across. This is because latter paths will only be equivalent in length and thus not satisfy the invariant. However, in the case of using $\leq$ we define the shortest path to be the last shortest path we come across. This is because latter paths will be equivalent in length and thus trigger the invariant.

My point is that with $&lt;$ or $\leq$ we could essentially flip the ordering of nodes in the adjacency list of adjacency matrix to mimic what the other comparison operator would do.