**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

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P), A) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P, u(A)),\quad \varphi \mapsto u(\varphi ) \circ i_ P \]

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

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P_2, P_1) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P_2), v(P_1)),\quad \varphi \mapsto v(\varphi ) \]

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

\[ P_2 \to P_1 \to B \to 0 \]

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

\[ v(P_2) \to v(P_1) \to v(B) \to 0 \]

Then

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(B), A) & = \mathop{\mathrm{Ker}}(\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P_1), A) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(v(P_2), A)) \\ & = \mathop{\mathrm{Ker}}(\mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P_1, u(A)) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(P_2, u(A))) \\ & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(B, u(A)) \end{align*}

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

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