How about adding another object $W$ to your example with an automorphism $\delta:W\to W$ with $\delta^{-1}=\delta$ and arrows $\theta,\phi:X\to W$ with $\delta\circ\theta=\phi=\theta\circ\gamma$?

Then $\delta$ is null-homotopic but neither absorbed nor co-absorbed by any arrow, I think.