Since $f$ is a proper morphism we see that $f_*\mathcal{F}$ is a coherent $\mathcal{O}_Y$-module (Proposition \ref{proposition-proper-pushforward-coherent}). Thus the statement of the lemma makes sense. Consider the compositions $$ \gamma_n : \mathcal{G}_n \to f_*f^*\mathcal{G}_n \to f_*(\mathcal{F}/\mathcal{I}^n\mathcal{F}). $$ Here the first map is the adjunction map and the second is $f_*\beta_n$. We claim that there exists a unique $\alpha$ as in the lemma such that the compositions $$ \mathcal{G}_n \xrightarrow{\alpha_n} f_*\mathcal{F}/\mathcal{J}^nf_*\mathcal{F} \to f_*(\mathcal{F}/\mathcal{I}^n\mathcal{F}) $$ equal $\gamma_n$ for all $n$. Because of the uniqueness we may assume that $Y = \Spec(B)$ is affine. Let $J \subset B$ corresponds to the ideal $\mathcal{J}$. Set $$ M_n = H^0(X, \mathcal{F}/\mathcal{I}^n\mathcal{F}) \quad\text{and}\quad M = H^0(X, \mathcal{F}) $$ By Lemma \ref{lemma-ML-cohomology-powers-ideal} and Theorem \ref{theorem-formal-functions} the inverse limit of the modules $M_n$ equals the completion $M^\wedge = \lim M/J^nM$. Set $N_n = H^0(Y, \mathcal{G}_n)$ and $N = \lim N_n$. Via the equivalence of categories of Lemma \ref{lemma-inverse-systems-affine} the finite $B^\wedge$ modules $N$ and $M^\wedge$ correspond to $(\mathcal{G}_n)$ and $f_*\mathcal{F}^\wedge$. It follows from this that $\alpha$ has to be the morphism of $\textit{Coh}(Y, \mathcal{J})$ corresponding to the homomorphism $$ \lim \gamma_n : N = \lim_n N_n \longrightarrow \lim M_n = M^\wedge $$ of finite $B^\wedge$-modules. \medskip\noindent We still have to show that the kernel and cokernel of $\alpha$ are annihilated by a power of $\mathcal{K}$. Set $Y' = \Spec(B^\wedge)$ and $X' = Y' \times_Y X$. Let $\mathcal{K}'$, $\mathcal{J}'$, $\mathcal{G}'_n$ and $\mathcal{I}'$, $\mathcal{F}'$ be the pullback of $\mathcal{K}$, $\mathcal{J}$, $\mathcal{G}_n$ and $\mathcal{I}$, $\mathcal{F}$, to $Y'$ and $X'$. The projection morphism $f' : X' \to Y'$ is the base change of $f$ by $Y' \to Y$. Note that $Y' \to Y$ is a flat morphism of schemes as $B \to B^\wedge$ is flat by Algebra, Lemma \ref{algebra-lemma-completion-flat}. Hence $f'_*\mathcal{F}'$, resp.\ $f'_*(f')^*\mathcal{G}_n'$ is the pullback of $f_*\mathcal{F}$, resp.\ $f_*f^*\mathcal{G}_n$ to $Y'$ by Lemma \ref{lemma-flat-base-change-cohomology}. The uniqueness of our construction shows the pullback of $\alpha$ to $Y'$ is the corresponding map $\alpha'$ constructed for the situation on $Y'$. Moreover, to check that the kernel and cokernel of $\alpha$ are annihilated by $\mathcal{K}^t$ it suffices to check that the kernel and cokernel of $\alpha'$ are annihilated by $(\mathcal{K}')^t$. Namely, to see this we need to check this for kernels and cokernels of the maps $\alpha_n$ and $\alpha'_n$ (see Remark \ref{remark-inverse-systems-kernel-cokernel-annihilated-by}) and the ring map $B \to B^\wedge$ induces an equivalence of categories between modules annihilated by $J^n$ and $(J')^n$, see More on Algebra, Lemma \ref{more-algebra-lemma-neighbourhood-equivalence}. Thus we may assume $B$ is complete with respect to $J$. \medskip\noindent Assume $Y = \Spec(B)$ is affine, $\mathcal{J}$ corresponds to the ideal $J \subset B$, and $B$ is complete with respect to $J$. In this case $(\mathcal{G}_n)$ is in the essential image of the functor $\textit{Coh}(\mathcal{O}_Y) \to \textit{Coh}(Y, \mathcal{J})$. Say $\mathcal{G}$ is a coherent $\mathcal{O}_Y$-module such that $(\mathcal{G}_n) = \mathcal{G}^\wedge$. Note that $f^*(\mathcal{G}^\wedge) = (f^*\mathcal{G})^\wedge$. Hence Lemma \ref{lemma-fully-faithful} tells us that $\beta$ comes from an isomorphism $b : f^*\mathcal{G} \to \mathcal{F}$ and $\alpha$ is the completion functor applied to $$ \mathcal{G} \to f_*f^*\mathcal{G} \cong f_*\mathcal{F} $$ Hence we are trying to verify that the kernel and cokernel of the adjunction map $c : \mathcal{G} \to f_*f^*\mathcal{G}$ are annihilated by a power of $\mathcal{K}$. However, since the restriction $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is an isomorphism we see that $c|_V$ is an isomorphism. Thus the coherent sheaves $\Ker(c)$ and $\Coker(c)$ are supported on $V(\mathcal{K})$ hence are annihilated by a power of $\mathcal{K}$ (Lemma \ref{lemma-power-ideal-kills-sheaf}) as desired.