Say $RF$ is defined at $X, Y$ with values $A, B$. Let $RF(f) : A \to B$ be the induced morphism, see Lemma \ref{lemma-derived-functor}. We may choose a distinguished triangle $(A, B, C, RF(f), b, c)$ in $\mathcal{D}'$. We claim that $C$ is a value of $RF$ at $Z$. \medskip\noindent To see this pick $s : X \to X'$ in $S$ such that there exists a morphism $\alpha : A \to F(X')$ as in Categories, Definition \ref{categories-definition-essentially-constant-diagram}. We may choose a commutative diagram $$ \xymatrix{ X \ar[d]_f \ar[r]_s & X' \ar[d]^{f'} \\ Y \ar[r]^{s'} & Y' } $$ with $s' \in S$ by MS2. Using that $Y/S$ is filtered we can (after replacing $s'$ by some $s'' : Y \to Y''$ in $S$) assume that there exists a morphism $\beta : B \to F(Y')$ as in Categories, Definition \ref{categories-definition-essentially-constant-diagram}. Picture $$ \xymatrix{ A \ar[d]_{RF(f)} \ar[r]_-\alpha & F(X') \ar[r] \ar[d]^{F(f')} & A \ar[d]^{RF(f)} \\ B \ar[r]^-\beta & F(Y') \ar[r] & B } $$ It may not be true that the left square commutes, but the outer and right squares commute. The assumption that the ind-object $\{F(Y')\}_{s' : Y' \to Y}$ is essentially constant means that there exists a $s'' : Y \to Y''$ in $S$ and a morphism $h : Y' \to Y''$ such that $s'' = h \circ s'$ and such that $F(h)$ equal to $F(Y') \to B \to F(Y') \to F(Y'')$. Hence after replacing $Y'$ by $Y''$ and $\beta$ by $F(h) \circ \beta$ the diagram will commute (by direct computation with arrows). \medskip\noindent Using MS6 choose a morphism of triangles $$ (s, s', s'') : (X, Y, Z, f, g, h) \longrightarrow (X', Y', Z', f', g', h') $$ with $s'' \in S$. By TR3 choose a morphism of triangles $$ (\alpha, \beta, \gamma) : (A, B, C, RF(f), b, c) \longrightarrow (F(X'), F(Y'), F(Z'), F(f'), F(g'), F(h')) $$ \medskip\noindent By Lemma \ref{lemma-derived-inverts} it suffices to prove that $RF(Z')$ is defined and has value $C$. Consider the category $\mathcal{I}$ of Lemma \ref{lemma-limit-triangles} of triangles $$ \mathcal{I} = \{(t, t', t'') : (X', Y', Z', f', g', h') \to (X'', Y'', Z'', f'', g'', h'') \mid (t, t', t'') \in S\} $$ To show that the system $F(Z'')$ is essentially constant over the category $Z'/S$ is equivalent to showing that the system of $F(Z'')$ is essentially constant over $\mathcal{I}$ because $\mathcal{I} \to Z'/S$ is cofinal, see Categories, Lemma \ref{categories-lemma-cofinal-essentially-constant} (cofinality is proven in Lemma \ref{lemma-limit-triangles}). For any object $W$ in $\mathcal{D}'$ we consider the diagram $$ \xymatrix{ \colim_\mathcal{I} \Mor_{\mathcal{D}'}(W, F(X'')) & \Mor_{\mathcal{D}'}(W, A) \ar[l] \\ \colim_\mathcal{I} \Mor_{\mathcal{D}'}(W, F(Y'')) \ar[u] & \Mor_{\mathcal{D}'}(W, B) \ar[u] \ar[l] \\ \colim_\mathcal{I} \Mor_{\mathcal{D}'}(W, F(Z'')) \ar[u] & \Mor_{\mathcal{D}'}(W, C) \ar[u] \ar[l] \\ \colim_\mathcal{I} \Mor_{\mathcal{D}'}(W, F(X''[1])) \ar[u] & \Mor_{\mathcal{D}'}(W, A[1]) \ar[u] \ar[l] \\ \colim_\mathcal{I} \Mor_{\mathcal{D}'}(W, F(Y''[1])) \ar[u] & \Mor_{\mathcal{D}'}(W, B[1]) \ar[u] \ar[l] } $$ where the horizontal arrows are given by composing with $(\alpha, \beta, \gamma)$. Since filtered colimits are exact (Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}) the left column is an exact sequence. Thus the $5$ lemma (Homology, Lemma \ref{homology-lemma-five-lemma}) tells us the $$ \colim_\mathcal{I} \Mor_{\mathcal{D}'}(W, F(Z'')) \longrightarrow \Mor_{\mathcal{D}'}(W, C) $$ is bijective. Choose an object $(t, t', t'') : (X', Y', Z') \to (X'', Y'', Z'')$ of $\mathcal{I}$. Applying what we just showed to $W = F(Z'')$ and the element $\text{id}_{F(X'')}$ of the colimit we find a unique morphism $c_{(X'', Y'', Z'')} : F(Z'') \to C$ such that for some $(X'', Y'', Z'') \to (X''', Y''', Z'')$ in $\mathcal{I}$ $$ F(Z'') \xrightarrow{c_{(X'', Y'', Z'')}} C \xrightarrow{\gamma} F(Z') \to F(Z'') \to F(Z''') \quad\text{equals}\quad F(Z'') \to F(Z''') $$ The family of morphisms $c_{(X'', Y'', Z'')}$ form an element $c$ of $\lim_\mathcal{I} \Mor_{\mathcal{D}'}(F(Z''), C)$ by uniqueness (computation omitted). Finally, we show that $\colim_\mathcal{I} F(Z'') = C$ via the morphisms $c_{(X'', Y'', Z'')}$ which will finish the proof by  Categories, Lemma \ref{categories-lemma-characterize-essentially-constant-ind}. Namely, let $W$ be an object of $\mathcal{D}'$ and let $d_{(X'', Y'', Z'')} : F(Z'') \to W$ be a family of maps corresponding to an element of $\lim_\mathcal{I} \Mor_{\mathcal{D}'}(F(Z''), W)$. If $d_{(X', Y', Z')} \circ \gamma = 0$, then for every object $(X'', Y'', Z'')$ of $\mathcal{I}$ the morphism $d_{(X'', Y'', Z'')}$ is zero by the existence of $c_{(X'', Y'', Z'')}$ and the morphism $(X'', Y'', Z'') \to (X''', Y''', Z'')$ in $\mathcal{I}$ satisfying the displayed equality above. Hence the map $$ \lim_\mathcal{I} \Mor_{\mathcal{D}'}(F(Z''), W) \longrightarrow \Mor_{\mathcal{D}'}(C, W) $$ (coming from precomposing by $\gamma$) is injective. However, it is also surjective because the element $c$ gives a left inverse. We conclude that $C$ is the colimit by Categories, Remark \ref{categories-remark-limit-colim}.