The idea will be to first extend $F$ to a sufficiently large collection of affine schemes over $S$ and then use the Zariski sheaf property to extend to all schemes. \medskip\noindent Suppose that $V$ is an affine scheme over $S$ whose structure morphism $V \to S$ factors through some affine open $U \subset S$. In this case we can write $$ V = \lim V_i $$ as a cofiltered limit with $V_i \to U$ of finite presentation and $V_i$ affine. See Algebra, Lemma \ref{algebra-lemma-ring-colimit-fp}. By conditions (1) and (2) we may replace our $V_i$ by objects of $\mathcal{C}$. Observe that $V_i \to S$ is locally of finite presentation (if $S$ is quasi-separated, then these morphisms are actually of finite presentation). Then we set $$ F'(V) = \colim F(V_i) $$ Actually, we can give a more canonical expression, namely $$ F'(V) = \colim_{V \to V'} F(V') $$ where the colimit is over the category of morphisms $V \to V'$ over $S$ where $V'$ is an object of $\mathcal{C}$ whose structure morphism $V' \to S$ is locally of finite presentation. The reason this is the same as the first formula is that by Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation} our inverse system $V_i$ is cofinal in this category! Finally, note that if $V$ were an object of $\mathcal{C}$, then $F'(V) = F(V)$ by assumption (b). \medskip\noindent The second formula turns $F'$ into a contravariant functor on the category formed by affine schemes $V$ over $S$ whose structure morphism factors through an affine open of $S$. Let $V$ be such an affine scheme over $S$ and suppose that $V = \bigcup_{k = 1, \ldots, n} V_k$ is a finite open covering by affines. Then it makes sense to ask if the sheaf condition holds for $F'$ and this open covering. This is true and easy to show: write $V = \lim V_i$ as in the previous paragraph. By Limits, Lemma \ref{limits-lemma-descend-opens} for all sufficiently large $i$ we can find affine opens $V_{i, k} \subset V_i$ compatible with transition maps pulling back to $V_k$ in $V$. Thus $$ F'(V_k) = \colim F(V_{i, k}) \quad\text{and}\quad F'(V_k \cap V_l) = \colim F(V_{i, k} \cap V_{i, l}) $$ Strictly speaking in these formulas we need to replace $V_{i, k}$ and $V_{i, k} \cap V_{i, l}$ by isomorphic affine objects of $\mathcal{C}$ before applying the functor $F$. Since $I$ is directed the colimits pass through equalizers. Hence the sheaf condition (b) for $F$ and the Zariski coverings $\{V_{i, k} \to V_i\}$ implies the sheaf condition for $F'$ and this covering. \medskip\noindent Let $X$ be a general scheme over $S$. Let $\mathcal{B}_X$ denote the collection of affine opens of $X$ whose structure morphism to $S$ maps into an affine open of $S$. It is clear that $\mathcal{B}_X$ is a basis for the topology of $X$. By the result of the previous paragraph and Sheaves, Lemma \ref{sheaves-lemma-cofinal-systems-coverings-standard-case} we see that $F'$ is a sheaf on $\mathcal{B}_X$. Hence $F'$ restricted to $\mathcal{B}_X$ extends uniquely to a sheaf $F'_X$ on $X$, see Sheaves, Lemma \ref{sheaves-lemma-extend-off-basis}. If $X$ is an object of $\mathcal{C}$ then we have a canonical identification $F'_X(X) = F(X)$ by the agreement of $F'$ and $F$ on the objects for which they are both defined and the fact that $F$ satisfies the sheaf condition for Zariski coverings. \medskip\noindent Let $f : X \to Y$ be a morphism of schemes over $S$. We get a unique $f$-map from $F'_Y$ to $F'_X$ compatible with the maps $F'(V) \to F'(U)$ for all $U \in \mathcal{B}_X$ and $V \in \mathcal{B}_Y$ with $f(U) \subset V$, see Sheaves, Lemma \ref{sheaves-lemma-f-map-basis-above-and-below-structures}. We omit the verification that these maps compose correctly given morphisms $X \to Y \to Z$ of schemes over $S$. We also omit the verification that if $f$ is a morphism of $\mathcal{C}$, then the induced map $F'_Y(Y) \to F'_X(X)$ is the same as the map $F(Y) \to F(X)$ via the identifications $F'_X(X) = F(X)$ and $F'_Y(Y) = F(Y)$ above. In this way we see that the desired extension of $F$ is the functor which sends $X/S$ to $F'_X(X)$. \medskip\noindent Property (a) for the functor $X \mapsto F'_X(X)$ is almost immediate from the construction; we omit the details. Suppose that $X = \lim_{i \in I} X_i$ is a directed limit of affine schemes over $S$. We have to show that $$ F'_X(X) = \colim_{i \in I} F'_{X_i}(X_i) $$ First assume that there is some $i \in I$ such that $X_i \to S$ factors through an affine open $U \subset S$. Then $F'$ is defined on $X$ and on $X_{i'}$ for $i' \geq i$ and we see that $F'_{X_{i'}}(X_{i'}) = F'(X_{i'})$ for $i' \geq i$ and $F'_X(X) = F'(X)$. In this case every arrow $X \to V$ with $V$ locally of finite presentation over $S$ factors as $X \to X_{i'} \to V$ for some $i' \geq i$, see Limits, Proposition \ref{limits-proposition-characterize-locally-finite-presentation}. Thus we have \begin{align*} F'_X(X) & = F'(X)  \\ & = \colim_{X \to V} F(V) \\ & = \colim_{i' \geq i} \colim_{X_{i'} \to V} F(V) \\ & = \colim_{i' \geq i} F'(X_{i'}) \\ & = \colim_{i' \geq i} F'_{X_{i'}}(X_{i'}) \\ & = \colim_{i' \in I} F'_{X_{i'}}(X_{i'}) \end{align*} as desired. Finally, in general we pick any $i \in I$ and we choose a finite affine open covering $V_i = V_{i, 1} \cup \ldots \cup V_{i, n}$ such that $V_{i, k} \to S$ factors through an affine open of $S$. Let $V_k \subset V$ and $V_{i', k}$ for $i' \geq i$ be the inverse images of $V_{i, k}$. By the previous case we see that $$ F'_{V_k}(V_k) = \colim_{i' \geq i} F'_{V_{i', k}}(V_{i', k}) $$ and $$ F'_{V_k \cap V_l}(V_k \cap V_l) = \colim_{i' \geq i} F'_{V_{i', k} \cap V_{i', l}}(V_{i', k} \cap V_{i', l}) $$ By the sheaf property and exactness of filtered colimits we find that $F'_X(X) = \colim_{i \in I} F'_{X_i}(X_i)$ also in this case. This finishes the proof of property (b) and hence finishes the proof of the lemma.