Ext can be computed from any projective resolution of the first variable

Choose $a_M:Q_{\bullet }(M)\to P_{\bullet }(M)$ lifting $1_M$. A reverse comparison is its homotopy inverse since both composites lift the identity and [F2] compares them with identity chain maps.

Precomposition gives $a_M^{\ast }:\mathrm{Hom}(P_{\bullet }(M),N)\to \mathrm{Hom}(Q_{\bullet }(M),N)$. A homotopy $a-b=dh+hd$ induces the cochain homotopy $s^n(f)=f\circ h_{n-1}$, with $s^0=0$. Thus $H^n(a_M^{\ast })$ is a choice-independent isomorphism. Its source is ${\mathrm{Ext}}_P^n(M,N)$ by Ext via a projective resolution of the first variable, hence balanced Ext by The balanced Ext bifunctor.

For $u:M^{\mathrm{\prime }}\to M$, choose lifts $P(u)$ and $Q(u)$ by [F1]. Precomposition defines their cohomology actions independently of the lifts by [F2]. Identities and composition follow because composites lift the object composites. The maps $P(u)a_{M^{\mathrm{\prime }}}$ and $a_{M}Q(u)$ both lift $u$, so [F2] makes them homotopic. Precomposition and cohomology therefore give the required contravariant naturality square in $M$.

Postcomposition by $v:N\to N^{\mathrm{\prime }}$ commutes exactly with precomposition by $a_M$. This gives naturality in $N$ and hence both variables. For a single supplied resolution $Q_{\bullet }(M)$, steps 1.1–2.1 already give the canonical objectwise isomorphism. No simultaneous class-wide choice of comparison maps is required.