Is there a name for a kind of map that simultaneously transforms structures and wffs?
The motivation for this kind of map is the existence of various systems like free logic and plural logic. It's easy to prove that these things are equivalent to $\mathsf{FOL}$ but I'm curious if there's a mathematical object that can be used as a concrete witness of this equivalence.
There are basically two approaches to interpreting equality in FOL. One is to say that $=$ is a congruence with respect to every function symbol and is transitive, reflexive, and symmetric. We also constrain $=$ so that no two equal things can be distinguished by any relation symbol. In this case, we use the term normal model to identify models where $=$ lines up with true equality. The other approach is to define $=$ directly in the semantics.
These two approaches are completely equivalent. Here's a way to talk about their equivalence. I'll use $\equiv$ to represent equality-as-a-congruence just so the map isn't trivial.
Let $\mu,\nu$ be maps from structures+wffs to structures+wffs, in a sense made precise below.
Let $\mathsf{FOL}$ refer to first-order logic with equality and $\mathsf{WFOL}$ refer to first-order logic without equality (i.e. with equality as a congruence).
Let $M$ be a $\mathsf{FOL}$-model. I define $\mu(M)$ as follows.
$\mu(M)_D$ is defined as $M_D$.
$[\![f]\!]_{\mu(M)}$ where $f$ is a function symbol is defined as $[\![f]\!]_M$.
$[\![R]\!]_{\mu(M)}$ where $R$ is a relation symbol is defined as $[\![R]\!]_M$.
$[\![\equiv]\!]_{\mu(M)}$ is $\{(a,a): a \in M\}$.
$\mu(\varphi)$ is just $\varphi$ with $=$ replaced with $\equiv$.
Similarly, let $N$ be a $\mathsf{WFOL}$-model.
$\nu(N)_D$ is $\nu(N)_D$ modded out by $[\![\equiv]\!]$.
$[\![f]\!]_{\nu(N)}$ is determined by taking a representative from each equivalence class given as an argument, determining the result using $[\![f]\!]_N$, and then determining the equivalence class that it belongs to.
$[\![R]\!]_{\nu(N)}$ is determined by taking a representative from each equivalence class and seeing whether that representative is in $[\![R]\!]_N$.
$\nu(\varphi)$ is $\varphi$ but with $\equiv$ replaced with $=$.
So, both maps $\mu,\nu$ preserve and reflect truth ($\models$).
Also, the kind of thing that $\mu$ and $\nu$ are is interesting, since their properties are preserved by composition.
We can also consider "endomorphism"-like maps, such as the one that sends each structure to an ultrapower and leaves the language alone.
There's also a "truth-functional" map that just sends all $M, \varphi$ pairs such that $M \models \varphi$ to the same place and all $M, \varphi$ paris such that $M \not\models \varphi$ to a different place.
There might be a way to rule out "truth-functional" maps since they inspect the structure of $M$ and $\varphi$ fairly deeply; I'm not sure.
I'm curious if there's a real notion out there that this kind of map corresponds to.
This kind of thing seems similar to a functor with sentences being like objects and models like morphisms, since functors simultaneously map objects and arrows. However, each model is only associated with one sentence in this setting, so it's not a perfect fit.