Let $L(X)$ be loop space of $X$ which is path connected surface (and its universal cover is contractible and $\pi_n=0$ for $n >1$ this details i'm giving as i'm working on such surface.), let $a$ and $b$ be two equivalence class of $L(X)$.
In class $a$ we identify the loops with its based loop homotopy class and name it $L(X)/a$. That is two loops are in same class if if they have based loop homotopy between them.
In class $b$ we identify loops with its conjugacy classes and all its based loop homotopy classes and name it $L(X)/b$. That is two loops $\alpha$ and $\beta$ are in same class if we can find loop $\gamma$ such that $[\gamma*\alpha*\bar\gamma]=[\beta]$ all this loops have same based point.
So we have $f : L(X) \rightarrow L(X)/a$
and we have $g : L(X) \rightarrow L(X)/b$
So my question is will this two maps be Fibration?