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:

Writing $\psi'$ for the path going backwards along an arrow $\psi$:

$$\delta\sim\theta'\theta\delta\sim\theta'\gamma\theta\sim\theta'\operatorname{id}_X\theta\sim\theta'\theta\sim\operatorname{id}_W$$

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:

Writing $\psi'$ for the path going backwards along an arrow $\psi$:

$$\delta\sim\delta\theta\theta'\sim\theta\gamma\theta'\sim\theta\operatorname{id}_X\theta'\sim\theta\theta'\sim\operatorname{id}_W$$

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.

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:

Writing $\psi'$ for the path going backwards along an arrow $\psi$:

$$\delta\sim\theta'\theta\delta\sim\theta'\gamma\theta\sim\theta'\operatorname{id}_X\theta\sim\theta'\theta\sim\operatorname{id}_W$$