Lemma 87.2.1. Choose a category of schemes \mathit{Sch}_\alpha as in Sets, Lemma 3.9.2. Given a formal scheme \mathfrak X let
h_\mathfrak X : (\mathit{Sch}_\alpha )^{opp} \longrightarrow \textit{Sets},\quad h_\mathfrak X(S) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Formal Schemes}}(S, \mathfrak X)
be its functor of points. Then we have
\mathop{\mathrm{Mor}}\nolimits _{\textit{Formal Schemes}}(\mathfrak X, \mathfrak Y) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathit{Sch}_\alpha )}(h_\mathfrak X, h_\mathfrak Y)
provided the size of \mathfrak X is not too large.
Proof.
First we observe that h_\mathfrak X satisfies the sheaf property for the Zariski topology for any formal scheme \mathfrak X (see Schemes, Definition 26.15.3). 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 26.15.3). 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 26.15.3). 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.
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 \mathop{\mathrm{Spec}}(A/I_\lambda ) for all \lambda we find an open subscheme U_\lambda \subset \mathop{\mathrm{Spec}}(A/I_\lambda ). However, for I_\lambda \subset I_\mu the morphism \mathop{\mathrm{Spec}}(A/I_\lambda ) \to \mathop{\mathrm{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.
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 : \mathop{\mathrm{Spec}}(A/I_\lambda ) \to \mathfrak X maps to a morphism \varphi (i_\lambda ) : \mathop{\mathrm{Spec}}(A/I_\lambda ) \to \mathfrak Y. By (87.2.0.1) 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 = \mathop{\mathrm{lim}}\nolimits \chi _\lambda : B \longrightarrow \mathop{\mathrm{lim}}\nolimits 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 (87.2.0.1). We omit the verification that this construction is the inverse to the map of the lemma in this case.
Set theoretic remarks. To make this work on the given category of schemes \mathit{Sch}_\alpha we just have to make sure all the schemes used in the proof above are isomorphic to objects of \mathit{Sch}_\alpha . In fact, a careful analysis shows that it suffices if the schemes \mathop{\mathrm{Spec}}(A/I_\lambda ) occurring above are isomorphic to objects of \mathit{Sch}_\alpha . For this it certainly suffices to assume the size of \mathfrak X is at most the size of a scheme contained in \mathit{Sch}_\alpha .
\square
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