The Stacks project

Proof. By the Yoneda lemma (Categories, Lemma 4.3.5) the object $Q$ of $\mathcal{A}$ corresponding to $P$ is defined up to unique isomorphism by the formula $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(Q, A) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P, u(A))$. Let us write $Q = v(P)$. Denote $i_ P : P \to u(v(P))$ the map corresponding to $\text{id}_{v(P)}$ in $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P), v(P))$. Functoriality in (2) implies that the bijection is given by

For any pair of elements $P_1, P_2 \in \mathcal{P}$ there is a canonical map

which is characterized by the rule $u(v(\varphi )) \circ i_{P_2} = i_{P_1} \circ \varphi $ in $\mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P_2, u(v(P_1)))$. Note that $\varphi \mapsto v(\varphi )$ is compatible with composition; this can be seen directly from the characterization. Hence $P \mapsto v(P)$ is a functor from the full subcategory of $\mathcal{B}$ whose objects are the elements of $\mathcal{P}$.

Given an arbitrary object $B$ of $\mathcal{B}$ choose an exact sequence

which is possible by assumption (1). Define $v(B)$ to be the object of $\mathcal{A}$ fitting into the exact sequence

Then

Hence we see that we may take $\mathcal{P} = \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$, i.e., we see that $v$ is everywhere defined. $\square$