PlanetPhysics/Section Functor

Essential data
Let us consider an abelian category $$\mathcal{C}$$ which is locally small and a dense subcategory $$\mathcal{A}$$ of $$\mathcal{C}$$, with $$T: \mathcal{C} \to \mathcal{C}/\mathcal{A}$$ being the canonical functor. Moreover, let us assume that $$T$$ has a right adjoint denoted by $$S$$ such that one has the following functorial isomorphism, or natural equivalence:

$$Hom_{\mathcal{C}}(X, S(Y)) \cong Hom_{\mathcal{C} / \mathcal{A}}$$.

The right adjoint functor $$S: \mathcal{C}/ \mathcal{A} \to \mathcal{C}$$ of $$T$$-- which is specified by the essential data above-- is called a section functor.

Note: the category $$\mathcal{A}$$ is defined as a localizing subcategory.

Reference cited.