Standard or good names for relations between maps

Question

I am looking for standard (or good) names for identities between maps of the form $$\tag{1}\label{1} s \circ \pi = \rho \circ s, \quad s: A\to Z, \pi: A\to A, \rho: Z\to Z $$ or, more generally, $$ s(\pi(a_1,\dots,a_k)) = \rho(s(a_1),\dots,s(a_k))\quad s: A\to Z, \pi: A^k\to A, \rho: Z^k\to Z $$ and $$\tag{2}\label{2} s \circ \phi \circ s = \psi, \quad s: A\to Z, \phi: Z\to A, \psi: Z\to A. $$

I lean towards calling \eqref{1} an intertwining relation, but I am not sure what to call \eqref{2}. I'd be grateful for any suggestion.

Background:

This question came up in two different projects: on the one hand, findstat discovers such identities automatically, see for example http://www.findstat.org/MapsDatabase/Mp00111 or http://www.findstat.org/MapsDatabase/Mp00064 (click to show experimental identities). Since there are often very many (conjectural, based on numerical evidence only!) identities, it is useful to somehow classify them.

On the other hand, together with some collegues, I put together a bijectionist's toolkit, which is a tool that produces data to find a bijection or statistic given various constraints. In this setting, the map $s: A\to Z$ is sought for, and $\pi$, $\rho$, $\phi$ and $\psi$ are supplied.

Answer 1

In dynamics for \eqref{1}, one says that $s$ is a semi-conjugacy between $\pi$ and $\rho$, or that $\pi$ and $\rho$ are semi-conjugate with semi-conjugacy $s$. This terminology probably comes from the fact that if $s$ is invertible, then $\pi=s^{-1}\circ\rho\circ s$, so $\pi$ and $\rho$ are conjugate to one another. The semi-conjugacy relation is very useful, since it implies that the iterates of $\pi$ and $\rho$ are also semi-conjugate, by the same semi-conjugacy. One might also say that $\pi$ is fibered over $\rho$ (by $s$), since $\pi$ maps fibers to fibers in the commutative diagram $$\require{AMScd}\begin{CD} A @>\pi>> A \\ @VVsV @VVsV \\ Z @>\rho>> Z\end{CD}$$