Before giving the proofs, we explain the meaning of (c). Suppose we have an additional cartesian square $$ \xymatrix{ X'' \ar[d]_{f''} \ar[r]_{h'} & X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y'' \ar[r]^h & Y' \ar[r]^g & Y } $$ tacked onto our given diagram. If (a) holds, then there is a canonical map $\gamma : h^*R^df'_*\mathcal{F}' \to R^df''_*(h')^*\mathcal{F}'$. Namely, $\gamma$ is the map on degree $d$ cohomology sheaves induced by the composition $$ Lh^*Rf'_*\mathcal{F}' \longrightarrow Rf''_*L(h')^*\mathcal{F}' \longrightarrow Rf''_*(h')^*\mathcal{F}' $$ Here the first arrow is the base change map (Cohomology, Remark \ref{cohomology-remark-base-change}) and the second arrow complex from the canonical map $L(g')^*\mathcal{F} \to (g')^*\mathcal{F}$. Similarly, since $Rf'_*\mathcal{F}$ has no nonzero cohomology sheaves in degrees $> d$ by (a) we have $H^d(Lh^*Rf_*\mathcal{F}') = h^*R^df_*\mathcal{F}$. The content of (c) is that $\gamma$ is an isomorphism. \medskip\noindent Having said this, we can check (a), (b), and (c) locally on $Y'$ and $Y''$. Suppose that $V \subset Y$ is a quasi-compact open subscheme. Then we claim (1) and (2) hold for $f|_{f^{-1}(V)} : f^{-1}(V) \to V$. Namely, (1) is immediate and (2) follows because any quasi-coherent module on $f^{-1}(V)$ is the restriction of a quasi-coherent module on $X$ (Properties, Lemma \ref{properties-lemma-extend-trivial}) and formation of higher direct images commutes with restriction to opens. Thus we may also work locally on $Y$. In other words, we may assume $Y''$, $Y'$, and $Y$ are affine schemes. \medskip\noindent Proof of (a) when $Y'$ and $Y$ are affine. In this case the morphisms $g$ and $g'$ are affine. Thus $g_* = Rg_*$ and $g'_* = Rg'_*$ (Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}) and $g_*$ is identified with the restriction functor on modules (Schemes, Lemma \ref{schemes-lemma-widetilde-pullback}). Then $$ g_*(R^if'_*\mathcal{F}') = H^i(Rg_*Rf'_*\mathcal{F}') = H^i(Rf_*Rg'_*\mathcal{F}') = H^i(Rf_*g'_*\mathcal{F}') = Rf^i_*g'_*\mathcal{F}' $$ which is zero by assumption (2). Hence (a) by our description of $g_*$. \medskip\noindent Proof of (b) when $Y'$ is affine, say $Y' = \Spec(R')$. By part (a) we have $H^{d + 1}(X', \mathcal{F}') = 0$ for any quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$, see Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images-application}. Consider the functor $F$ on $R'$-modules defined by the rule $$ F(M) = H^d(X', \mathcal{F}' \otimes_{\mathcal{O}_{X'}} (f')^*\widetilde{M}) $$ By Cohomology, Lemma \ref{cohomology-lemma-quasi-separated-cohomology-colimit} this functor commutes with direct sums (this is where we use that $X$ and hence $X'$ is quasi-compact and quasi-separated). On the other hand, if $M_1 \to M_2 \to M_3 \to 0$ is an exact sequence, then $$ \mathcal{F}' \otimes_{\mathcal{O}_{X'}} (f')^*\widetilde{M}_1 \to \mathcal{F}' \otimes_{\mathcal{O}_{X'}} (f')^*\widetilde{M}_2 \to \mathcal{F}' \otimes_{\mathcal{O}_{X'}} (f')^*\widetilde{M}_3 \to 0 $$ is an exact sequence of quasi-coherent modules on $X'$ and by the vanishing of higher cohomology given above we get an exact sequence $$ F(M_1) \to F(M_2) \to F(M_3) \to 0 $$ In other words, $F$ is right exact. Any right exact $R'$-linear functor $F : \text{Mod}_{R'} \to \text{Mod}_{R'}$ which commutes with direct sums is given by tensoring with an $R'$-module (omitted; left as exercise for the reader). Thus we obtain $F(M) = H^d(X', \mathcal{F}') \otimes_{R'} M$. Since $R^d(f')_*\mathcal{F}'$ and $R^d(f')_*(\mathcal{F}' \otimes_{\mathcal{O}_{X'}} (f')^*\widetilde{M})$ are quasi-coherent (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherence-higher-direct-images}), the fact that $F(M) = H^d(X', \mathcal{F}') \otimes_{R'} M$ translates into the statement given in (b). \medskip\noindent Proof of (c) when $Y'' \to Y' \to Y$ are morphisms of affine schemes. Say $Y'' = \Spec(R'')$ and $Y' = \Spec(R')$. Then we see that $R^df''_*(h')^*\mathcal{F}'$ is the quasi-coherent module on $Y'$ associated to the $R''$-module $H^d(X'', (h')^*\mathcal{F}')$. Now $h' : X'' \to X'$ is affine hence $H^d(X'', (h')^*\mathcal{F}') = H^d(X, h'_*(h')^*\mathcal{F}')$ by the already used Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-cohomology}. We have $$ h'_*(h')^*\mathcal{F}' = \mathcal{F}' \otimes_{\mathcal{O}_{X'}} (f')^*\widetilde{R''} $$ as the reader sees by checking on an affine open covering. Thus $H^d(X'', (h')^*\mathcal{F}') = H^d(X', \mathcal{F}') \otimes_{R'} R''$ by part (b) applied to $f'$ and the proof is complete.