Assume $\mathcal{F}$ is a sheaf. Condition (1) holds because a Zariski covering is a h covering, see Lemma \ref{lemma-zariski-h}. Condition (2) holds because for $f$ as in (2) we have that $\{X \to Y\}$ is an fppf covering (this is clear) and hence an h covering, see Lemma \ref{lemma-zariski-h}. Condition (3) holds by Lemma \ref{lemma-blow-up-square-h}. \medskip\noindent Conversely, assume $\mathcal{F}$ satisfies (1), (2), and (3). We will prove $\mathcal{F}$ is a sheaf by applying Lemma \ref{lemma-characterize-sheaf-h}. Consider a surjective, finitely presented, proper morphism $f : X \to Y$ in $(\Sch/S)_h$ with $Y$ affine. It suffices to show that $\mathcal{F}(Y)$ is the equalizer of the two maps $\mathcal{F}(X) \to \mathcal{F}(X \times_Y X)$. \medskip\noindent First, assume that $f : X \to Y$ is in addition a closed immersion (in other words, $f$ is a thickening). Then the blow up of $Y$ in $X$ is the empty scheme and this produces an almost blow up square consisting with $\emptyset, \emptyset, X, Y$ at the vertices (compare with the second proof of Lemma \ref{lemma-thickening-h}). Hence we see that condition (3) tells us that $$ \xymatrix{ \mathcal{F}(\emptyset) & \mathcal{F}(\emptyset) \ar[l] \\ \mathcal{F}(X) \ar[u] & \mathcal{F}(Y) \ar[u] \ar[l] } $$ is cartesian in the category of sets. Since $\mathcal{F}$ is a sheaf for the Zariski topology, we see that $\mathcal{F}(\emptyset)$ is a singleton. Hence we see that $\mathcal{F}(X) = \mathcal{F}(Y)$. \medskip\noindent Interlude A: let $T \to T'$ be a morphism of $(\Sch/S)_h$ which is a thickening and of finite presentation. Then $\mathcal{F}(T') \to \mathcal{F}(T)$ is bijective. Namely, choose an affine open covering $T' = \bigcup T'_i$ and let $T_i = T \times_{T'} T'_i$ be the corresponding affine opens of $T$. Then we have $\mathcal{F}(T'_i) \to \mathcal{F}(T_i)$ is bijective for all $i$ by the result of the previous paragraph. Using the Zariski sheaf property we see that $\mathcal{F}(T') \to \mathcal{F}(T)$ is injective. Repeating the argument we find that it is bijective. Minor details omitted. \medskip\noindent Interlude B: consider an almost blow up square (\ref{equation-almost-blow-up-square}) in the category $(\Sch/S)_h$. Then we claim the diagram $$ \xymatrix{ \mathcal{F}(E) & \mathcal{F}(X') \ar[l] \\ \mathcal{F}(Z) \ar[u] & \mathcal{F}(X) \ar[u] \ar[l] } $$ is cartesian in the category of sets. This is a consequence of condition (3) as follows by choosing an affine open covering of $X$ and arguing as in Interlude A. We omit the details. \medskip\noindent Next, let $f : X \to Y$ be a surjective, finitely presented, proper morphism in $(\Sch/S)_h$ with $Y$ affine. Choose a generic flatness stratification $$ Y \supset Y_0 \supset Y_1 \supset \ldots \supset Y_t = \emptyset $$ as in Lemma \ref{lemma-generic-flatness-stratification} for $f : X \to Y$. We are going to use all the properties of the stratification without further mention. Set $X_0 = X \times_Y Y_0$. By the Interlude B we have $\mathcal{F}(Y_0) = \mathcal{F}(Y)$, $\mathcal{F}(X_0) = \mathcal{F}(X)$, and $\mathcal{F}(X_0 \times_{Y_0} X_0) = \mathcal{F}(X \times_Y X)$. \medskip\noindent We are going to prove the result by induction on $t$. If $t = 1$ then $X_0 \to Y_0$ is surjective, proper, flat, and of finite presentation and we see that the result holds by property (2). For $t > 1$ we may replace $Y$ by $Y_0$ and $X$ by $X_0$ (see above) and assume $Y = Y_0$. \medskip\noindent Consider the quasi-compact open subscheme $V = Y \setminus Y_1 = Y_0 \setminus Y_1$. Choose a diagram $$ \xymatrix{ E \ar[ddd] \ar[rd] & & & D \ar[lll] \ar[ddd] \ar[ld] \\ & Y' \ar[d] & X' \ar[l] \ar[d] \\ & Y & X \ar[l] \\ Z \ar[ru] & & & T \ar[lll] \ar[lu] } $$ as in Lemma \ref{lemma-flat-after-almost-blowing-up} for $f : X \to Y \supset V$. Then $f' : X' \to Y'$ is flat and of finite presentation. Also $f'$ is proper (use Morphisms, Lemmas \ref{morphisms-lemma-composition-proper} and \ref{morphisms-lemma-image-proper-scheme-closed} to see this). Thus the image $W = f'(X') \subset Y'$ is an open (Morphisms, Lemma \ref{morphisms-lemma-fppf-open}) and closed subscheme of $Y'$. Observe that $Y' \setminus E$ is contained in $W$. By Lemma \ref{lemma-shrink-almost-blow-up} this means we may replace $Y'$ by $W$ in the above diagram. In other words, we may and do assume $f'$ is surjective. At this point we know that $$ \vcenter{ \xymatrix{ \mathcal{F}(E) & \mathcal{F}(Y') \ar[l] \\ \mathcal{F}(Z) \ar[u] & \mathcal{F}(Y) \ar[u] \ar[l] } } \quad\text{and}\quad \vcenter{ \xymatrix{ \mathcal{F}(D) & \mathcal{F}(X') \ar[l] \\ \mathcal{F}(T) \ar[u] & \mathcal{F}(X) \ar[u] \ar[l] } } $$ are cartesian by Interlude B. Note that $Z \cap Y_1 \to Z$ is a thickening of finite presentation (as $Z$ is set theoretically contained in $Y_1$ as a closed subscheme of $Y$ disjoint from $V$). Thus we obtain a filtration $$ Z \supset Z \cap Y_1 \supset Z \cap Y_2 \subset \ldots \subset Z \cap Y_t = \emptyset $$ as above for the restriction $T = Z \times_Y X \to Z$ of $f$ to $T$. Thus by induction hypothesis we find that $\mathcal{F}(Z) \to \mathcal{F}(T)$ is an injective map of sets whose image is the equalizer of the two maps $\mathcal{F}(T) \to \mathcal{F}(T \times_Z T)$. \medskip\noindent Let $s \in \mathcal{F}(X)$ be in the equalizer of the two maps $\mathcal{F}(X) \to \mathcal{F}(X \times_Y X)$. By the above we see that the restriction $s|_T$ comes from a unique element $t \in \mathcal{F}(Z)$ and similarly that the restriction $s|_{X'}$ comes from a unique element $t' \in \mathcal{F}(Y')$. Chasing sections using the restriction maps for $\mathcal{F}$ corresponding to the arrows in the huge commutative diagram above the reader finds that $t$ and $t'$ restrict to the same element of $\mathcal{F}(E)$ because they restrict to the same element of $\mathcal{F}(D)$ and we have (2); here we use that $D \to E$ is surjective, flat, proper, and of finite presentation as the restriction of $X' \to Y'$. Thus by the first of the two cartesian squares displayed above we get a unique section $u \in \mathcal{F}(Y)$ restricting to $t$ and $t'$ on $Z$ and $Y'$. To see that $u$ restrict to $s$ on $X$ use the second diagram.