Let $\mathcal{K}$ be a locally presentable category. I recall that a cylinder $C:\mathcal{K}\to \mathcal{K}$ is by definition equipped with two natural maps $\gamma_X:X\sqcup X\to CX$ and $\sigma_X:CX\to X$ such that the composite $\sigma_X\gamma_X$ is the codiagonal of $X$. And that a cocylinder $P:\mathcal{K}\to \mathcal{K}$ is by definition equipped with two natural maps $\xi_X:X\to PX$ and $\pi_X:PX\to X\times X$ such that the composite $\pi_X\xi_X$ is the diagonal of $X$.
Let $C:\mathcal{K}\to \mathcal{K}$ be a cylinder of $\mathcal{K}$ which has a right adjoint $P:\mathcal{K}\to \mathcal{K}$. Let $\eta$ be the unit of the adjunction and let $\epsilon$ be the counit of the adjunction. Then this functor $P$ can be equipped with a structure of cocylinder as follows. The map $\xi_X:X\to PX$ is the composite map $$X\stackrel{\eta_X}\to PCX \stackrel{P(\sigma_X)}\to PX$$ and the map $\pi_X:PX\to X\times X$ is the image of the identity of $PX$ by the composite maps $$\mathcal{K}(PX,PX) \cong \mathcal{K}(CPX,X) \to \mathcal{K}(PX \sqcup PX,X) \cong \mathcal{K}(PX,X\times X).$$
Do you know an example with another possible structure of cocylinder on $P$ (an exotic structure) ?
Or another way to formulate the question:
Consider a cylinder functor $C:\mathcal{K}\to \mathcal{K}$ and a cocylinder functor $P:\mathcal{K}\to \mathcal{K}$ which are adjoint as functors, are the maps $\xi_X:X\to PX$ and $\xi_X:X\to PX$ necessarily associated to each other by the adjunction ?
Motivation: I am just curious, I am not hiding any exotic path functor in my pocket.
