1 Introduction
1.1 Over category
If \(a : F \vdash G\) is an adjunction between \(F : C \to D\) and \(G : D \to C\) and \(X : C\), then there is an adjunction between \(F / X : C / X \to D / F(X)\) and \(G / X : D / F(X) \to C / X\).
See https://ncatlab.org/nlab/show/sliced+adjoint+functors+–+section.
Let \(J\) be a shape (i.e. a category). Let \(\widetilde J\) denote the category which is the same as \(J\), but has an extra object \(*\) which is terminal. If \(F : C \to D\) is a functor preserving limits of shape \(\widetilde J\), then the obvious functor \(C / X \to D / F(X)\) preserves limits of shape \(J\).
Extend a functor \(K\colon J \to C / X\) to a functor \(\widetilde K\colon \widetilde J \to C\), by letting \(\widetilde K (*) = X\).
If \(F : C \to D\) is a full functor between cartesian-monoidal categories, then \(F / X : C / X \hom D / F(X)\) has the same essential image as \(F\).
Transfer all diagrams.