Consider a pair of embeddings $f,g : S^1 \rightarrow M$ in the category of topological manifolds with boundary. Assume $M$ is a surface with boundary, and that $f$ and $g$ land in the boundary, and thus have a coqualizer. I get the feeling that the coequalizer of $f$ and $g$ should be isomorphic to the coequalizer of $f$ and $g \circ h$, for any orientation-preserving homeomorphism $h:S^1 \rightarrow S^1$.
Question. Is this correct, and if so, how does one prove it?
Also, is there a standard name for this result?
Feel free to use smooth manifolds if you prefer.