Hint : use the fact that there is a fixed (infinite) bound on $|G(i)|, i\in \mathrm{Ob}(\I)$, and so there's at most a set of $\I$-indexed families of groups on which $G(i)$ surjects, and so at most a set of natural transformations from $G$ to those, and that each natural $G\implies \Delta(H)$ factors through such a natural transformation.
The problem is that you cannot choose a domain and codomain for such a putative adjunction consistently and simultaneously. The statement that we have is that the category of presheaves on $C$ is the free cocomplete category on $C$ when $C$ is small. However, the forgetful functor from cocomplete categories to categories does not land in small categories-every cocomplete category which is not a preorder is large.
So, you might want an improved adjunction between large categories and cocomplete categories. However, the category of presheaves on a large category is even larger than large! What this means depends on your foundations. If we work with universes so that a small category is $U_1$-small, then presheaves on a small category are small with respect to the next biggest universe $U_2$. Now presheaves on a $U_2$-small category only have the appropriate universal property if we can them to be presheaves of $U_2$-small sets, and such presheaves are not $U_2$-small.
So the presheaves-forgetful functor pair cannot form an adjunction, because the desired left adjoint moves us up a universe every time we apply it. Thus in particular we cannot get around this straightforwardly by using universes.
There are a couple of partial solutions to this problem. The simplest is to regard the formation of the presheaf category as a relative left adjoint to the forgetful functor from cocomplete categories to (possibly) large categories. In other words, it behaves like a left adjoint, but is only partially defined. This is a rephrasing of the theorem you quote-the formation of presheaves behaves like a left adjoint when its input is small.
A more technical approach is to ask the question: even if the formation of presheaves cannot be left adjoint to the forgetful functor from cocomplete categories to large categories, does this forgetful functor have any left adjoint at all? In fact it does; for instance, it satisfies a 2-categorical version of the general adjoint functor theorem. This left adjoint sends a category $K$ to the subcategory of presheaves on $K$ formed by the colimits of small diagrams of representable presheaves. However, this category of small presheaves is not nearly as well behaved as the presheaf category. It needn't even be a topos in general.
So to summarize, this is a real issue which cannot be eradicated by any level of generous assumptions on the foundations. It's the go-to example of why size issues cannot be completely ignored in category theory.
Best Answer
First let us recall the fact that every presheaf is a colimit of representables. Just Googling this sentence should already give you plenty of proofs, but just to link two: Wikipedia and another question on math.se. That is, given a presheaf $P$ in $\mathcal{C}^*$ (keeping you notation for the category of presheaves), we have that $P = \operatorname{colim}_{i \in I} y(C_i)$ for some objects $C_i$ in $\mathcal{C}$. Here $y$ denotes the Yoneda embedding (I will use $y'$ for the Yoneda embedding $y': \mathcal{D} \to \mathcal{D}^*$, just to make clear which one is used when).
Now the proof proceeds by abstract nonsense (essentially this is working out the direction Derek tried to point towards in the comments). The proof is essentially a long chain of natural isomorphisms (or simply actual equalities). It may be a good exercise to stop at each step and see if you can do the rest by yourself. For this reason I have broken up this chain at every step, but if you wish you can just read all the centred math/text and paste it together in one long chain of isomorphisms.
Let $P$ be a presheaf in $\mathcal{C}^*$ and let $D$ be some object in $\mathcal{D}$. Then by definition $$ \operatorname{Hom}(P, S(D)) = \operatorname{Hom}(P, F^* y'(D)). $$ Since every presheaf is a colimit of representables we have $$ \operatorname{Hom}(P, F^* y'(D)) \cong \operatorname{Hom}(\operatorname{colim}_{i \in I} y(C_i), F^* y'(D)). $$ Then using that Hom-sets turn colimits in their first argument into limits we have $$ \operatorname{Hom}(\operatorname{colim}_{i \in I} y(C_i), F^* y'(D)) \cong \lim_{i \in I} \operatorname{Hom}(y(C_i), F^* y'(D)). $$ By the Yoneda-lemma we then find $$ \lim_{i \in I} \operatorname{Hom}(y(C_i), F^* y'(D)) \cong \lim_{i \in I} F^* y'(D)(C_i). $$ By definition of $F^*$ this gives us $$ \lim_{i \in I} F^* y'(D)(C_i) = \lim_{i \in I} y'(D)(F(C_i)). $$ By the definition of the Yoneda-embedding we then have $$ \lim_{i \in I} y'(D)(F(C_i)) = \lim_{i \in I} \operatorname{Hom}(F(C_i), D). $$ Once more using that Hom-sets convert limits into colimits in their first argument (here we use that $\mathcal{D}$ is cocomplete), we obtain $$ \lim_{i \in I} \operatorname{Hom}(F(C_i), D) \cong \operatorname{Hom}(\operatorname{colim}_{i \in I} F(C_i), D). $$ This then gives us the desired description for our left adjoint: send a presheaf $P \cong \operatorname{colim}_{i \in I} y(C_i)$ to $\operatorname{colim}_{i \in I} F(C_i)$.
Of course, to really prove that this gives you a functor, you would still need to check that this works for arrows as well. Here is an idea for what to do. In the same style as above, so you can stop at any time to try to do the rest on your own.
Let $P = \operatorname{colim}_{i \in I} y(C_i)$ and $Q = \operatorname{colim}_{j \in J} y(C_j)$. The idea is that, given an arrow $P \to Q$, to get an arrow $\operatorname{colim}_{i \in I} F(C_i) \to \operatorname{colim}_{j \in J} F(C_j)$, we will want to use the universal property of the colimit $\operatorname{colim}_{i \in I} F(C_i)$. That is, we will try to make $\operatorname{colim}_{j \in J} F(C_j)$ into a cocone for the diagram $(F(C_i))_{i \in I}$. Again, we will make some identifications, starting with $$ \operatorname{Hom}(P, Q) = \operatorname{Hom}(\operatorname{colim}_{i \in I} y(C_i), \operatorname{colim}_{j \in J} y(C_j)). $$ Pulling the colimit out of the first argument gives us $$ \operatorname{Hom}(\operatorname{colim}_{i \in I} y(C_i), \operatorname{colim}_{j \in J} y(C_j)) \cong \lim_{i \in I} \operatorname{Hom}(y(C_i), \operatorname{colim}_{j \in J} y(C_j)). $$ Then by the Yoneda lemma we get $$ \lim_{i \in I} \operatorname{Hom}(y(C_i), \operatorname{colim}_{j \in J} y(C_j)) \cong \lim_{i \in I} (\operatorname{colim}_{j \in J} y(C_j))(C_i). $$ Using the fact that colimits in presheaf categories are calculated pointwise we then turn this into $$ \lim_{i \in I} (\operatorname{colim}_{j \in J} y(C_j))(C_i) = \lim_{i \in I} \operatorname{colim}_{j \in J} (y(C_j)(C_i)), $$ which by the definition of the Yoneda embedding becomes $$ \lim_{i \in I} \operatorname{colim}_{j \in J} (y(C_j)(C_i)) = \lim_{i \in I} \operatorname{colim}_{j \in J} \operatorname{Hom}(C_i, C_j). $$ So to sum up, we have $$ \operatorname{Hom}(P, Q) \cong \lim_{i \in I} \operatorname{colim}_{j \in J} \operatorname{Hom}(C_i, C_j). $$ An arrow $P \to Q$ thus corresponds to some tuple $([f_i])_{i \in I}$ of equivalence classes of arrows in $\mathcal{C}$. If you write out the definitions here, we have that arrows $f_i: C_i \to C_j$ and $f_i': C_i \to C_{j'}$ are in the same equivalence class if they factor through the diagram $(C_j)_{j \in J}$ in essentially the same way. That is, there is $j^*$, $a: C_j \to C_{j^*}$ and $b: C_{j'} \to C_{j^*}$ such that $a f_i = b f_i'$. This means precisely that if we have a cocone of the $(C_j)_{j \in J}$, then an equivalence class $[f_i]$ determines a unique arrow into the vertex of that cocone. In particular, if we take images under $F$ we get that each equivalence class $[f_i]$ determines a unique arrow $g_i: F(C_i) \to \operatorname{colim}_{j \in J} F(C_j)$ by composing $F(f_i)$ with the coprojection of $F(C_j)$ into the colimit.
So from the tuple $([f_i])_{i \in I}$ we then obtain a tuple of arrows $(g_i)_{i \in I}$ (remember, where $g_i: F(C_i) \to \operatorname{colim}_{j \in J} F(C_j)$). Writing out the definition of the limit we obtained that tuple from, we precisely get that $(g_i)_{i \in I}$ forms a cocone for the diagram $(F(C_i))_{i \in I}$.
So we used an arrow $P \to Q$ to make $\operatorname{colim}_{j \in J} F(C_j)$ into the vertex of a cocone of $(F(C_i))_{i \in I}$. Which means that we obtain an arrow $\operatorname{colim}_{i \in I} F(C_i) \to \operatorname{colim}_{j \in J} F(C_j)$, which is the arrow we will send our original $P \to Q$ to.