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A construction that I've been finding all over the place in studying the category of NF (Quine's New Foundations) sets and functions is a situation like the following: there's a functor $T:\mathbf{C}\to \mathbf{Set_{NF}}$ and a functor $F:\mathbf{D}\to \mathbf{Set_{NF}}$ such that for all $c\in\mathbf{C}$ there exists a universal arrow $\langle \sigma_c, d_c\rangle$ from $Tc$ to $F$ (i.e. $(Tc\downarrow F)$ has an initial object). In the wild, $T$ is usually an endofunctor, and $F$ is a functor you would expect to have a left adjoint in standard set theories. The effect is that some universal construction is only really defined on objects in the image of $T$.

If I haven't screwed up, it looks like this means one can define a functor $H:\mathbf{C}\to \mathbf{D}$ such that the above universal arrows form the components of a natural transformation $\sigma: T\to FH$. Moreover, it looks like $\langle \sigma, H\rangle$ form a universal arrow from $T$ to $F^{\mathbf{C}}:\mathbf{D^C}\to\mathbf{Set_{NF}^C}$; sort of like a Kan extension turned on its head.

Question Time: Besides "are my conclusions correct," I'm wondering if this is an instance of something well-studied, and if there are any good references on that something.

Apologies if this has a painfully obvious answer, or if the question has come out unclear.

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What does NF mean? – Martin Brandenburg Jul 5 '14 at 19:27
Sorry, Quine's New Foundations set theory. – Malice Vidrine Jul 5 '14 at 20:18
Do you have an example of when this happens that isn't specific to NF? – Qiaochu Yuan Jul 5 '14 at 23:00
@Qiaochu: Well, when the $T$ in the above example is the identity, it's just an adjunction. Besides that, I cannot think of any examples where $T$ is something non-trivial. – Malice Vidrine Jul 6 '14 at 0:07
(Well, no non-trivial examples outside NF, that is.) – Malice Vidrine Jul 6 '14 at 0:31
up vote 1 down vote accepted

What I have described above, somewhat idiosyncratically, is a relative adjoint. Specifically, $F$ is a functor with a $T$-relative left adjoint.

Moreover the "Kan extension turned on its head" is a Kan lift.

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