First we observe that $h_\mathfrak X$ satisfies the sheaf property for the Zariski topology for any formal scheme $\mathfrak X$ (see Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}). This follows from the local nature of morphisms in the category of formal schemes. Also, for an open immersion $\mathfrak V \to \mathfrak W$ of formal schemes, the corresponding transformation of functors $h_\mathfrak V \to h_\mathfrak W$ is injective and representable by open immersions (see Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}). Choose an open covering $\mathfrak X = \bigcup \mathfrak U_i$ of a formal scheme by affine formal schemes $\mathfrak U_i$. Then the collection of functors $h_{\mathfrak U_i}$ covers $h_\mathfrak X$ (see Schemes, Definition \ref{schemes-definition-representable-by-open-immersions}). Finally, note that $$ h_{\mathfrak U_i} \times_{h_\mathfrak X} h_{\mathfrak U_j} = h_{\mathfrak U_i \cap \mathfrak U_j} $$ Hence in order to give a map $h_\mathfrak X \to h_\mathfrak Y$ is equivalent to giving a family of maps $h_{\mathfrak U_i} \to h_\mathfrak Y$ which agree on overlaps. Thus we can reduce the bijectivity (resp.\ injectivity) of the map of the lemma to bijectivity (resp.\ injectivity) for the pairs $(\mathfrak U_i, \mathfrak Y)$ and injectivity (resp.\ nothing) for $(\mathfrak U_i \cap \mathfrak U_j, \mathfrak Y)$. In this way we reduce to the case where $\mathfrak X$ is an affine formal scheme. Say $\mathfrak X = \text{Spf}(A)$ for some admissible topological ring $A$. Also, choose a fundamental system of ideals of definition $I_\lambda \subset A$. \medskip\noindent We can also localize on $\mathfrak Y$. Namely, suppose that $\mathfrak V \subset \mathfrak Y$ is an open formal subscheme and $\varphi : h_\mathfrak X \to h_\mathfrak Y$. Then $$ h_\mathfrak V \times_{h_\mathfrak Y, \varphi} h_\mathfrak X \to h_\mathfrak X $$ is representable by open immersions. Pulling back to $\Spec(A/I_\lambda)$ for all $\lambda$ we find an open subscheme $U_\lambda \subset \Spec(A/I_\lambda)$. However, for $I_\lambda \subset I_\mu$ the morphism $\Spec(A/I_\lambda) \to \Spec(A/I_\mu)$ pulls back $U_\mu$ to $U_\lambda$. Thus these glue to give an open formal subscheme $\mathfrak U \subset \mathfrak X$. A straightforward argument (omitted) shows that $$ h_\mathfrak U = h_\mathfrak V \times_{h_\mathfrak Y} h_\mathfrak X $$ In this way we see that given an open covering $\mathfrak Y = \bigcup \mathfrak V_j$ and a transformation of functors $\varphi :  h_\mathfrak X \to h_\mathfrak Y$ we obtain a corresponding open covering of $\mathfrak X$. Since $\mathfrak X$ is affine, we can refine this covering by a finite open covering $\mathfrak X = \mathfrak U_1 \cup \ldots \cup \mathfrak U_n$ by affine formal subschemes. In other words, for each $i$ there is a $j$ and a map $\varphi_i : h_{\mathfrak U_i} \to h_{\mathfrak V_j}$ such that $$ \xymatrix{ h_{\mathfrak U_i} \ar[r]_{\varphi_i} \ar[d] & h_{\mathfrak V_j} \ar[d] \\ h_{\mathfrak X} \ar[r]^\varphi & h_\mathfrak Y } $$ commutes. With a few additional arguments (which we omit) this implies that it suffices to prove the bijectivity of the lemma in case both $\mathfrak X$ and $\mathfrak Y$ are affine formal schemes. \medskip\noindent Assume $\mathfrak X$ and $\mathfrak Y$ are affine formal schemes. Say $\mathfrak X = \text{Spf}(A)$ and $\mathfrak Y = \text{Spf}(B)$. Let $\varphi : h_\mathfrak X \to h_\mathfrak Y$ be a transformation of functors. Let $I_\lambda \subset A$ be a fundamental system of ideals of definition. The canonical inclusion morphism $i_\lambda : \Spec(A/I_\lambda) \to \mathfrak X$ maps to a morphism $\varphi(i_\lambda) : \Spec(A/I_\lambda) \to \mathfrak Y$. By (\ref{equation-morphisms-affine-formal-schemes}) this corresponds to a continuous map $\chi_\lambda : B \to A/I_\lambda$. Since $\varphi$ is a transformation of functors it follows that for $I_\lambda \subset I_\mu$ the composition $B \to A/I_\lambda \to A/I_\mu$ is equal to $\chi_\mu$. In other words we obtain a ring map $$ \chi = \lim \chi_\lambda : B \longrightarrow \lim A/I_\lambda = A $$ This is a continuous homomorphism because the inverse image of $I_\lambda$ is open for all $\lambda$ (as $A/I_\lambda$ has the discrete topology and $\chi_\lambda$ is continuous). Thus we obtain a morphism $\text{Spf}(\chi) : \mathfrak X \to \mathfrak Y$ by (\ref{equation-morphisms-affine-formal-schemes}). We omit the verification that this construction is the inverse to the map of the lemma in this case. \medskip\noindent Set theoretic remarks. To make this work on the given category of schemes $\Sch_\alpha$ we just have to make sure all the schemes used in the proof above are isomorphic to objects of $\Sch_\alpha$. In fact, a careful analysis shows that it suffices if the schemes $\Spec(A/I_\lambda)$ occurring above are isomorphic to objects of $\Sch_\alpha$. For this it certainly suffices to assume the size of $\mathfrak X$ is at most the size of a scheme contained in $\Sch_\alpha$.