Let $S$ be a scheme. Let $\mathcal{U} = \{\varphi_i : U_i \to S\}_{i \in I}$ be a $\tau$-covering of $S$. Let $(X_i, \varphi_{ii'})$ be a descent datum relative to $\mathcal{U}$ and assume that each morphism $X_i \to U_i$ has property $\mathcal{P}$. We have to show there exists a scheme $X \to S$ such that $(X_i, \varphi_{ii'}) \cong (U_i \times_S X, can)$. \medskip\noindent Before we start the proof proper we remark that for any family of morphisms $\mathcal{V} : \{V_j \to S\}$ and any morphism of families $\mathcal{V} \to \mathcal{U}$, if we pullback the descent datum $(X_i, \varphi_{ii'})$ to a descent datum $(Y_j, \varphi_{jj'})$ over $\mathcal{V}$, then each of the morphisms $Y_j \to V_j$ has property $\mathcal{P}$ also. This is true because of assumption (1) that $\mathcal{P}$ is stable under any base change and the definition of pullback (see Definition \ref{definition-pullback-functor-family}). We will use this without further mention. \medskip\noindent First, let us prove the lemma when $S$ is affine. By Topologies, Lemma \ref{topologies-lemma-fpqc-affine}, \ref{topologies-lemma-fppf-affine}, \ref{topologies-lemma-etale-affine}, \ref{topologies-lemma-smooth-affine}, or \ref{topologies-lemma-syntomic-affine} there exists a standard $\tau$-covering $\mathcal{V} : \{V_j \to S\}_{j = 1, \ldots, m}$ which refines $\mathcal{U}$. The pullback functor $DD(\mathcal{U}) \to DD(\mathcal{V})$ between categories of descent data is fully faithful by Lemma \ref{lemma-refine-coverings-fully-faithful}. Hence it suffices to prove that the descent datum over the standard $\tau$-covering $\mathcal{V}$ is effective. By assumption (2) we see that $\coprod Y_j \to \coprod V_j$ has property $\mathcal{P}$. By Lemma \ref{lemma-family-is-one} this reduces us to the covering $\{\coprod_{j = 1, \ldots, m} V_j \to S\}$ for which we have assumed the result in assumption (3) of the lemma. Hence the lemma holds when $S$ is affine. \medskip\noindent Assume $S$ is general. Let $V \subset S$ be an affine open. By the properties of site the family $\mathcal{U}_V = \{V \times_S U_i \to V\}_{i \in I}$ is a $\tau$-covering of $V$. Denote $(X_i, \varphi_{ii'})_V$ the restriction (or pullback) of the given descent datum to $\mathcal{U}_V$. Hence by what we just saw we obtain a scheme $X_V$ over $V$ whose canonical descent datum with respect to $\mathcal{U}_V$ is isomorphic to $(X_i, \varphi_{ii'})_V$. Suppose that $V' \subset V$ is an affine open of $V$. Then both $X_{V'}$ and $V' \times_V X_V$ have canonical descent data isomorphic to $(X_i, \varphi_{ii'})_{V'}$. Hence, by Lemma \ref{lemma-refine-coverings-fully-faithful} again we obtain a canonical morphism $\rho^V_{V'} : X_{V'} \to X_V$ over $S$ which identifies $X_{V'}$ with the inverse image of $V'$ in $X_V$. We omit the verification that given affine opens $V'' \subset V' \subset V$ of $S$ we have $\rho^V_{V''} = \rho^V_{V'} \circ \rho^{V'}_{V''}$. \medskip\noindent By Constructions, Lemma \ref{constructions-lemma-relative-glueing} the data $(X_V, \rho^V_{V'})$ glue to a scheme $X \to S$. Moreover, we are given isomorphisms $V \times_S X \to X_V$ which recover the maps $\rho^V_{V'}$. Unwinding the construction of the schemes $X_V$ we obtain isomorphisms $$ V \times_S U_i \times_S X \longrightarrow V \times_S X_i $$ compatible with the maps $\varphi_{ii'}$ and compatible with restricting to smaller affine opens in $X$. This implies that the canonical descent datum on $U_i \times_S X$ is isomorphic to the given descent datum and we win.