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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:

  1. A category ${\bf Sig}$ called the category of $\mathbb{I}$-signatures.
  2. 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 $\Sigma$.
  3. 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.
  4. 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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    $\begingroup$ See ncatlab.org/nlab/show/dinatural+transformation , especially the second item in "Special Cases". $\endgroup$ Jul 16, 2022 at 0:38
  • $\begingroup$ @AndreasBlass Ah, thank you -- if you'd like to post that as an answer I'll accept. $\endgroup$
    – Alec Rhea
    Jul 16, 2022 at 2:27
  • $\begingroup$ I don't think the translation of the "satisfaction condition" into the form at the top of the question is correct. For example there may be sentences over $\Sigma'$ which are not in the image of the sentences over $\Sigma$, but are still satisfied by models of $\Sigma'$. Instead it should be an equality between two relations from models of $\Sigma'$ to sentences over $\Sigma$ (or vice versa), which is an ordinary natural transformation once we transpose one of the two functors. $\endgroup$ Jul 16, 2022 at 14:30
  • $\begingroup$ @ReidBarton The formulation at the top is not a translation, I copied it directly from Diaconescu's book Institution Independent Model Theory -- the suggestion that it can be reformulated as a natural transformation is my 'translation' of his formulation. Are you saying that you think his characterization is wrong but the natural transformation characterization fixes it? $\endgroup$
    – Alec Rhea
    Jul 16, 2022 at 20:49
  • $\begingroup$ What I mean is that the "commutative diagram" cannot be understood as an equality between a relation and a composition of three relations (two of which are functions). It must be understood as an equality between two compositions of two relations (in each of which, one relation is a function read in either the forwards or backwards direction). Therefore, it does not fit into the form found in "We can still talk about a 'natural transformation'..." $\endgroup$ Jul 17, 2022 at 1:37

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Promoting my comment to an answer, as suggested by OP: This is a special case of dinatural transformations. See, for example, https://ncatlab.org/nlab/show/dinatural+transformation , especially the second item in "Special Cases".

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