Suppose we have categories and functors as below

We can still talk about a 'natural transformation' $\alpha:F\Rightarrow G$ as consisting of a collection of arrows $\{\alpha_X:F(X)\to G(X)\}_{X\in{\bf Ob}_\mathcal{C}}$ such that

commutes for all arrows $f:X\to Y\in{\bf Hom}_\mathcal{C}$.

Have generalizations of natural transformations for coterminal functors with related domains been studied? What names do these notions go by?

This comes up (for example) studying institutions; recall that an institution $\mathbb{I}$ consists of:

- A category ${\bf Sig}$ called the
*category of $\mathbb{I}$-signatures*. - A functor $${\sf Sen}:{\bf Sig}\to{\bf Set}$$ assigning to each signature $\Sigma$ a set ${\sf Sen}(\Sigma)$ whose elements are called
*sentences over $\Sigma$*. - A functor $${\sf Mod}:{\bf Sig}^{op}\to{\bf Cat}$$ assigning to each signature $\Sigma$ a category ${\sf Mod}(\Sigma)$ whose objects are called
*$\Sigma$-models*and whose arrows are called*$\Sigma$-model homomorphisms*. - A collection of relations $\{\models_\Sigma:{\bf Ob}_{{\sf Mod}(\Sigma)}\to{\sf Sen}(\Sigma)\}_{\Sigma\in{\bf Ob}_{\bf Sig}}$ called
*$\Sigma$-satisfaction*, such that the following diagram commutes for all $\varphi:\Sigma\to\Sigma'\in{\bf Hom}_{\bf Sig}$. The requirement that these diagrams commute is called the*satisfaction condition*for $\mathbb{I}$.

Recall that any indexed category $P:\mathcal{C}\to\mathfrak{Cat}$ gives rise to an 'object part presheaf' $${\bf Ob}_P:\mathcal{C}\to{\bf Set}$$ $$X\mapsto{\bf Ob}_{P(X)},$$ $$f:X\to Y\longmapsto P(f)_0:{\bf Ob}_{P(X)}\to{\bf Ob}_{P(Y)}.$$ Recall further that any set-valued functor $F:\mathcal{C}\to{\bf Set}$ can be trivially extended to a relation-valued functor $F^{\leftrightarrow}:\mathcal{C}\to{\bf Rel}$ by postcomposing with the canonical inclusion ${\bf Set}\hookrightarrow{\bf Rel}$. Then the satisfaction condition is just naturality for a 'natural transformation' $$\models\;:{\bf Ob}_{\sf Mod}^{\leftrightarrow}\Rightarrow{\sf Sen}^{\leftrightarrow}.$$ This situation is obviously leveraging the fact that ${\bf Ob}_\mathcal{C}={\bf Ob}_{\mathcal{C}^{op}}$ and ${\bf Hom}_\mathcal{C}\cong{\bf Hom}_{\mathcal{C}^{op}}$, and it's pretty easy to generalize further to "natural transformations" between coterminal functors with equivalent domains, but I'm almost sure this is already generalized by something in the literature. Extranatural transformations and dinatural transformations didn't seem to fit the bill on the face of it, but I haven't really worked with either notion and it's certainly possible that I'm just not seeing how to interpret these transformations as appropriately extra/dinatural. Any pointers or references are appreciated.

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