Adjoints, naturality, exactness, small Yoneda lemma

Proof: For f ∈ Hom(X,A), i ◦ f = 0 implies (i ◦ f)(x) = 0 for all x ∈ X, and then f(x) = 0 for all x since i is injective. Thus, Hom(X,A)→ Hom(X,B) is injective, giving exactness at the left joint. Since q ◦ i = 0, any f ∈ Hom(X,A) is mapped to 0 ∈ Hom(X,C) by f → q ◦ i ◦ f . That is, the image of i ◦ − is contained in the kernel of q ◦ −. On the other hand, when g ∈ Hom(X,B) is mapped to q ◦ g = 0 in Hom(X,C), we have g(X) ⊂ ker q = Imi Since i is injective, it is an isomorphism to its image, so there is an inverse i−1 : i(A)→ A. Since g(X) ⊂ Imi we can define f = i−1 ◦ g ∈ Hom(X,A) [1] A functor F : C −→ D of categories whose hom-sets are abelian groups additive when the map on morphisms Hom(A,B) → Hom(FA,FB) given by F is a homomorphism of abelian groups. This also entails F (A ⊕ B) ≈ FA⊕ FB. These isomorphisms are required to be natural.