This question is a follow-up to Are there infinitely many L-rigs? which is already pretty convoluted.

Define the evaluation morphism at a complex number $s$ as $\epsilon_{s}:F\mapsto F(s)$ where $F$ if a map from $\mathbb{C}$ to itself. Say an L-rig $\mathcal{L}$ is a topological L-rig if any automorphism $\varphi$ thereof commutes with $\epsilon_{s}$ for all complex number $s$. Does it make the automorphism group of $\mathcal{L}$ of order at most $2$?