$\mathcal{C}$, $\mathcal{D}$ be abelian categories. $\mathcal{C}$ is cocomplete, and has set $\mathcal{P}$ of projective generators. $F$, $G\colon \mathcal{C} \to \mathcal{D}$ are additive functors and $F$ is cocontinuous. Given a natural transformation $\eta\colon F\vert_\mathcal{P} \Rightarrow G\vert_\mathcal{P}$, I want to extend it uniquely to a natural transformation $\tilde{\eta}\colon F \Rightarrow G$ where the generators are compact or $\mathcal{C}$ is AB5. Here an object $X$ is said compact if $Hom_\mathcal{C}(X, -)$ preserves coproducts.
Let us call an object free if it is a coproduct of generators(here $\mathcal{P}$), and finite free if it is a finite coproduct. Firstly for every free object $X$ fix a representation(since there may be many) $\bigoplus_{i \in I}P_i$ with injections $u_i\colon P_i \to X$ and define $\bar{\eta}_X$ as the unique morphism making following diagram commute(which exists by the cocontinuity of $F$). $\require{AMScd}$ $$\begin{CD} F(P_i) @>{F(u_i)}>> F(X)\\ @V{\eta_{P_i}}VV @V{\bar{\eta}_X}VV\\ G(P_i) @>{G(u_i)}>> G(X) \end{CD}$$ Let's first prove when the generators are compact. For every $P \in \mathcal{P}$ and $f\colon P \to X$, since $P$ is compact $f$ is factored as $h \circ g$ where $g\colon P \to \bigoplus_{i \in J}P_i$ for some finite $J \subset I$. Let $\theta\colon F(\bigoplus_{i \in J}P_i) \to G(\bigoplus_{i \in J}P_i)$ be the unique morphism making following diagram commute. $$\begin{CD} F(P_i) @>{F(u_i)}>> F(\bigoplus_{i \in J}P_i)\\ @V{\eta_{P_i}}VV @V{\theta}VV\\ G(P_i) @>{G(u_i)}>> G(\bigoplus_{i \in J}P_i) \end{CD}$$ Now one can prove that following two squares are commutative using that $F$ and $G$ are additive and thus preserves biproducts. $$\begin{CD} F(P) @>{F(g)}>> F(\bigoplus_{i \in J}P_i) @>{F(h)}>> F(X)\\ @V{\eta_{P}}VV @V{\theta}VV @V{\bar{\eta}_{X}}VV \\ G(P) @>{G(g)}>> G(\bigoplus_{i \in J}P_i) @>{G(h)}>> G(X) \end{CD}$$ It follows that $\bar{\eta}$ constructed so far is indeed natural(since free objects are coproducts of generators and $F$ is cocontinuous), independent of chosen representations, and extends $\eta$. We now extend this into arbitrary objects. Let $A$ be an object and fix a free resolution $F_1 \to F_0 \to A \to 0$. Define $\tilde{\eta}_A$ as the unique morphism making following diagram commute(which exists by the cocontinuity of $F$). $$\begin{CD} F(F_1) @>>> F(F_0) @>>> F(A) @>>> 0\\ @V{\bar{\eta}_{F_1}}VV @V{\bar{\eta}_{F_0}}VV @V{\tilde{\eta}_{A}}VV \\ G(F_1) @>>> G(F_0) @>>> G(A) \end{CD}$$ Let $B$ be another object with fixed resolution $F'_1 \to F'_0 \to B \to 0$. Let $f\colon A \to B$ be a morphism. Since free objects are projective we can construct following commutative diagram. $$\begin{CD} F_0 @>>> A @>>> 0\\ @VVV @V{f}VV \\ F'_0 @>>> B @>>> 0 \end{CD}$$ We know that the faces of the following cube commutes except the rightmost one. In fact, it also commutes because $F(F_0) \to F(A)$ is epic. This shows that not only $\tilde{\eta}$ is natural and independent of chosen resolutions, but also it extends $\bar{\eta}$, and a fortiori $\eta$.
On the other hand if $\mathcal{C}$ is AB5, above argument slightly modified shows that following diagram commutes where $X$ is free, $Y$ is finite free, $f\colon X \to Y$ is a morphism. $$\begin{CD} F(X) @>{F(f)}>> F(Y)\\ @V{\bar{\eta}_X}VV @V{\bar{\eta}_Y}VV\\ G(X) @>{G(f)}>> G(Y) \end{CD}$$ Again, above argument slightly modified shows that we can extend $\eta$ to $\dot{\eta}$ on finitely generated objects via fixed free resolutions $F_1 \to F_0 \to X \to 0$ where $F_0$ is finite free. Let $A$ be an object. Since $\mathcal{C}$ is Grothendieck thus well-powered, it is a direct limit of its (representatives of) finitely generated subobjects $(u_i\colon A_i \to A)_{i \in I}$. By cocontinuity of $F$, define $\tilde{\eta}_A$ as the unique morphism making following diagram commute(which exists by the cocontinuity of $F$). $$\begin{CD} F(A_i) @>{F(u_i)}>> F(A)\\ @V{\dot{\eta}_{A_i}}VV @V{\tilde{\eta}_A}VV\\ G(A_i) @>{G(u_i)}>> G(A) \end{CD}$$ It is evident that $\tilde{\eta}$ extends $\dot{\eta}$, and a fortiori $\eta$. Let $B$ be another object, and $f\colon A \to B$ be a morphism. Then the following diagram commutes, establishing the naturality of $\tilde{\eta}$. $$\begin{CD} F(A_i) @>>> F(f(A_i)) @>>> F(B)\\ @V{\dot{\eta}_{A_i}}VV @V{\dot{\eta}_{f(A_i)}}VV @V{\tilde{\eta}_B}VV\\ G(A_i) @>>> G(f(A_i)) @>>> G(B) \end{CD}$$
Every part of the construction was forced, so the extension is indeed unique.
Is this correct?