The Stacks project

Proof. Observe that $v$ is right exact as a left adjoint (Categories, Lemma 4.24.6). Combined with Lemma 12.7.2 this explains why (a) $\Leftrightarrow $ (b).

Assume (a). Let $I$ be an injective object of $\mathcal{A}$. Let $\varphi : N \to M$ be an injective map in $\mathcal{B}$ and let $\alpha : N \to uI$ be a morphism. By adjointness we get a morphism $\alpha : vN \to I$ and by assumption $v\varphi : vN \to vM$ is injective. Hence as $I$ is an injective object we get a morphism $\beta : vM \to I$ extending $\alpha $. By adjointness again this corresponds to a morphism $\beta : M \to uI$ extending $\alpha $. Hence (c) is true.

Assume $\mathcal{A}$ has enough injectives and (c) holds. Let $f : B \to B'$ be a monomorphism in $\mathcal{B}$, and let $A = \mathop{\mathrm{Ker}}(v(f))$. Choose a monomorphism $g : A \to I$ with $I$ injective (it exists by assumption). Then $g$ extends to $g' : v(B) \to I$, whence by adjunction a morphism $B \to u(I)$. Since $u(I)$ is injective, this morphism extends to $h : B' \to u(I)$, hence by adjunction a morphism $k : v(B') \to I$ extending $g'$. But then $k$ "extends” $g$, which forces $A = 0$ since $g$ was a monomorphism. Thus (a) is true. $\square$