= Solution
Let $F:\mathcal C\to\mathcal D$ be a <final functor>, and let $\lambda_c:GFc\to V$ be a <cocone under a diagram> $GF$. For $d\in\mathcal D$, choose an object $(c,a:d\to Fc)$ of the nonempty <comma category> $(d\downarrow F)$ and define
$$
\bar\lambda_d=\lambda_c\,G(a):Gd\to V.
$$
If $t:(c,a)\to(c',a')$ is a <morphism> in that comma category, then $Ft\,a=a'$, so the cocone equation gives $\lambda_cG(a)=\lambda_{c'}G(Ft)G(a)=\lambda_{c'}G(a')$. Equality also holds along a reversed arrow, and hence along any zigzag. Connectedness makes the definition independent of the chosen object. For $v:d\to d'$, use $(c,a:d'\to Fc)$ and $(c,av:d\to Fc)$ to obtain $\bar\lambda_d=\bar\lambda_{d'}Gv$. Thus $\bar\lambda$ is a cocone under $G$. Taking $(c,1_{Fc})$ shows that it extends $\lambda$.
Every extension must satisfy $\bar\lambda_d=\lambda_cG(a)$ by its cocone equation, proving uniqueness. The construction commutes with postcomposition by a <morphism> $V\to V'$. Hence restriction gives a natural bijection between the <sets> of cocones of $G$ and $GF$ with any fixed vertex. A universal cocone for $GF$ therefore extends to a universal cocone for $G$. In particular,
$$
\boxed{\operatorname{colim}_{\mathcal D}G\cong\operatorname{colim}_{\mathcal C}GF,}
$$
whenever the latter exists, and existence of all <colimits> of shape $\mathcal C$ implies existence of all colimits of shape $\mathcal D$.
For the factorization of an arbitrary <functor> $H:\mathcal C\to\mathcal E$, let
$$
P(e)=\pi_0(e\downarrow H),
$$
the <set> of <connected components of a category>, where components are defined by finite zigzags. Precomposition by $v:e\to e'$ gives a <functor> $(e'\downarrow H)\to(e\downarrow H)$ and therefore a map $P(v):P(e')\to P(e)$. Identity and composition laws make $P$ a <categorical presheaf>. This is a set-valued construction when the relevant comma categories are small or have only a set of components, as in the usual small-category convention for this factorization.
Take $\mathcal D$ to be the <category of elements> of this presheaf. Its objects are pairs $(e,S)$ with $S\in P(e)$; a morphism $(e,S)\to(e',S')$ is an arrow $v:e\to e'$ satisfying $P(v)(S')=S$. Define $G:\mathcal D\to\mathcal E$ by forgetting $S$. Given $v:e\to e'$ and an object $(e',S')$, the unique lift into that object has source $(e,P(v)(S'))$ and underlying arrow $v$. This proves that $G$ is a <discrete fibration>.
Define $F(c)=(Hc,[(c,1_{Hc})])$. For $t:c\to c'$, take $F(t)=Ht$. The arrow $t$ in $(Hc\downarrow H)$ joins $(c,1_{Hc})$ to $(c',Ht)$, so the required component equation holds. It follows that $F$ is a functor and $GF=H$.
Finally, an object of $((e,S)\downarrow F)$ is a pair $(c,v:e\to Hc)$ whose component in $(e\downarrow H)$ is $S$. Its morphisms are exactly the morphisms of that comma category between objects in $S$. Consequently $((e,S)\downarrow F)$ is precisely the full connected component $S$, and is nonempty and connected. Thus $F$ is final, and \b[every such functor factors as a final functor followed by a discrete fibration].
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