I wish I had the background to formalize them the way you have in your post.
In general, I get the importance of keeping in mind the relevant commutative diagram (often more than one) for whatever mapping you are working on, but it is not clear to me that you can get anything actionable out of them. Still, I do think that even if all your examples are in Set, CT is a good framework for thinking about them, without going down the abstraction levels.
Here is a couple of my somewhat related comments from awhile back: https://www.lesswrong.com/posts/NFMPftEhagCcHcmQJ/on-the-role-of-abstraction-in-mathematics-and-the-natural?commentId=GRp8PKJd64Tm5DhtA
https://www.lesswrong.com/posts/wuJpYLcMEBz4kcgAn/what-is-abstraction-1?commentId=qxnTe3TAYs2ZqzAQL
I wish I had the background to formalize them the way you have in your post.
In general, I get the importance of keeping in mind the relevant commutative diagram (often more than one) for whatever mapping you are working on, but it is not clear to me that you can get anything actionable out of them. Still, I do think that even if all your examples are in Set, CT is a good framework for thinking about them, without going down the abstraction levels.