= Solution
{c}
Let $\lambda_A:GF(A)\to Z$ be the given <cocone under a diagram>. For $B\in\mathcal D$, choose an object $(A,u:B\to FA)$ of the <comma category> $(B\downarrow F)$ and define
$$
\bar\lambda_B=\lambda_A G(u):GB\to Z.
$$
Such a choice exists because $F$ is a <final functor>. If $a:A\to A'$ gives a morphism $(A,u)\to(A',u')$, so $F(a)u=u'$, the cocone relation gives
$$
\lambda_{A'}G(u')=\lambda_{A'}GF(a)G(u)=\lambda_AG(u).
$$
The <connected category> condition makes the result independent of the choice along any zigzag.
For $v:B\to B'$, choose $(A,u:B'\to FA)$; then $(A,uv)$ is available for $B$, giving $\bar\lambda_B=\bar\lambda_{B'}G(v)$. Thus the maps form a <cocone under a diagram> on $G$. Choosing $(A,1_{FA})$ shows that restriction recovers $\lambda_A$. Any extension must satisfy $\bar\lambda_B=\bar\lambda_{FA}G(u)=\lambda_AG(u)$, proving uniqueness. This is <cocone extension along a final functor>.
If $GF$ has a <colimit>, extend its universal cocone in this way. Every cocone on $G$ restricts to one on $GF$, factors uniquely through that colimit, and then agrees with its extension by uniqueness. Hence the same vertex is a colimit of $G$. In particular,
$$
\boxed{\operatorname{colim}_{\mathcal D}G\cong\operatorname{colim}_{\mathcal C}GF,}
$$
and existence of all $\mathcal C$-shaped colimits in $\mathcal E$ implies existence of all $\mathcal D$-shaped colimits.
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