I am describing the question details, though the main question is short as below.

Let $O$ be the ring of integers of the finite extension $K$ of the $p$-adic field $\mathbb{Q}_p$. Let $R$ be a finite $\mathbb{Z}_p$-algebra. Let $\bar K$ be the algebraic closure of $K$ and $G_K:=\text{Gal}(\bar K/K)$ Then consider a *right exact* functor $$F: \mathscr{C} \to \mathscr{D}$$ between the abelian categories $\mathscr{C}$ and $\mathscr{D}$, where $\mathscr{C}$ is the category of finite $O \otimes_{\mathbb{Z}_p}R$-modules and $\mathscr{D}$ is the category of continuous representations on finite $R$-module (I think any abelian categories is ok for the below question).

**Question:** For a finite $O \otimes_{\mathbb{Z}_p}R$-module $M \in \mathscr{C}$ and any finite $R$-module $M'$, there is a natural isomorphism $$F(M) \otimes_RM' \overset{\simeq}{\longrightarrow} F(M \otimes_R M').$$
**Some hints is given**. It is asked to take a presentation of the module $M'$ and then to use the $5$-lemma to prove the above isomorphism. So I was trying the following way:
$$------------------------------------$$
**My efforts:**

For natural numbesr $m,n$, Take the finite presentation $R^{\otimes n} \to R^{\otimes m} \to M' \to 0$ of the finite $R$-module $M'$ and then do the following two things:

$(i)$ First take with tensor product $(O\otimes_{\mathbb{Z}_p} R)$-module $M$ and then apply the given right exact-functor $F$ to this finite presentation to obtain the exact sequence $$ F(M \otimes R^{\otimes n}) \to F(M \otimes R^{\otimes m}) \to F(M \otimes M') \to 0$$ $(ii)$ Take tensor product of the finite presentation $R^{\otimes n} \to R^{\otimes m} \to M' \to 0$ by $F(M)$ to obtain the exact sequence $$F(M) \otimes_R R^{\otimes n} \to F(M) \otimes_R R^{\otimes m} \to F(M) \otimes_R M' \to 0.$$ I think now from $(i)$ and $(ii)$, we have the following commutative diagram to apply $5$-lemma:

\begin{align} \matrix{ F(M \otimes_R R^{\otimes n}) &\to&F(M \otimes_R R^{\otimes m}) &\to&F(M \otimes_R M')&\to&0 \cr \downarrow f_1&&\downarrow f_2&&\downarrow {\color{red}{f_3}}&&\downarrow f_4&& \cr F(M) \otimes_R R^{\otimes n}&\to&F(M) \otimes_R R^{\otimes m}&\to& F(M) \otimes_R M' &\to& 0} \end{align} I think using $5$-lemma, we need to show ${\color{red}{f_3}}$ is an isomorphism to answer our question. I think we need to define $f_1, f_2$ in order to satisfy $5$-lemma criteria. But here we have $4$ columns and we can use $4$-lemma at best instead of $5$-lemma. How do get the $5$-th column according to the given hints.

Am I doing correct so far? Any guidance and help lease.