Suppose $S$ is a closed surface (assume $\chi(S)\leq 0$ if necessary). Let $p$ be a fixed point on the surface $S$, and $\beta$ be an oriented loop based at $p$.
Intuitively, I can put finger on $p$ and push $p$ along $\beta$, dragging the rest of surface along.
The statement I would like to confirm is: doing this will give me a homotopy from the identity $id_S:(S,p)\rightarrow (S,p)$ to a homeomorphism $f:(S,p)\rightarrow (S,p)$, and for any loop $\alpha$ based at $p$, $f_*([\alpha])=[\beta]\cdot[\alpha]\cdot[\beta]^{-1}$ where $f_*:\pi_1(S,p)\rightarrow \pi_1(S,p)$ is the induced homomorphism and $[\alpha],[\beta]$ are elements of $\pi(S,p)$ represented by $\alpha, \beta$ respectively.
Is this a true statement? If it is, could you give me a proof or a sketch of proof, or a reference for me to read? Thank you.
My question arose when I was reading the part of A Primer on Mapping Class Group introducing the Dehn-Nielsen-Baer theorem. It is claimed that there is a bijective correspondence between the set of free homotopy classes of unbased maps $S\rightarrow S$ to the set of conjugace classes of homomorphism $\pi_1(S)\rightarrow \pi_1(S)$. I know because $S$ is a $K(\pi_1(S),1)$ space, there is a bijection between the set of based homotopy classes of maps $(S,p)\rightarrow (S,p)$ and the set of all homomorphisms $\pi_1(S,p)\rightarrow \pi_1(S,p)$.