Homeomorphism from uniform convergence topology to compact-convergence

Question

Following-up on this post, let $C(\mathbb{R})$ and $C_0(\mathbb{R})$ be respectively equipped with the topology of compact-convergence and uniform convergence, respectively.

The Anderson-Kadec theorem guarantees that these spaces are indeed homeomorphic. Is it possible to explicitly describe this homeomorphism? Is it simply $$ f(x)\mapsto f(x)e^{-\|x\|}, $$ or something similar?