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.
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.