I know that $\mathsf{P}$ class is only defined for decision problems. Therefore, a problem like "Does there exist an $(s,t)$ path of length $k$ in the graph $G$?" is in $\mathsf{P}$. One can first find the shortest path from $s$ to $t$ in polynomial time and check if it is smaller than $k$ or not.
Now, consider the optimization version of this problem that we popularly call the shortest path problem: "Find a shortest $(s,t)$ path in the graph $G$". Since this is not a decision problem, is it wrong to say that "shortest path problem is in $\mathsf{P}$"? If so, in which complexity class does it belong to?