Construction of colimit via semi-final lift

Question

Let $U:J\rightarrow C$ be a diagram and $V:C\rightarrow D$ a functor such that there exists a colimit $(Y,\{g_\alpha:V(U(I_\alpha))\rightarrow Y\}_\alpha)$ of $V\circ U$ in $D$ and a semi-final lift $(X,\{f_\alpha:U(I_\alpha)\rightarrow X\}_\alpha,g:Y\rightarrow Y')$ of the $V$-structured sink $\{g_\alpha\}_\alpha$. Under which circumstances (if any) is $(X,\{f_\alpha\}_\alpha)$ colimit of $U$ in $C$?

Answer 1

I think it's enough if $V$ is faithful. The proof should work like follows:

(i) Observe that $(X,\{f_\alpha\}_\alpha)$ forms a cocone in $C$, here we use faithfulness of $V$ and the fact that $(Y',\{g\circ g_\alpha\}_\alpha)$ forms a cocone in $D$.

(ii) For another cocone $(X',\{f'_\alpha:U(I_\alpha)\rightarrow X'\}_\alpha)$ in $C$, it's image under $V$ is a cocone in $D$ so we can use the colimit property and find a unique $g'':Y\rightarrow V(X')$ in $D$ compatible with $\{V(f'_\alpha)\}_\alpha\}$ and $\{g_\alpha\}_\alpha$ so $(\{f'_\alpha\}_\alpha,g'')$ forms an element of $\{g_\alpha\}_\alpha/D$.

(iii) By the property of the semi-final lift, there is a unique morphism $f:X\rightarrow X'$ between the elements of $\{g_\alpha\}_\alpha/g$, which means $f$ is a morphism between the cocones in $C$ also. Conversly, any morphism $f'$ between the cocones induces a morphism between the same two elements in $\{g_\alpha\}_\alpha/D$ (via uniqueness of $g''$), hence equals $f$, which shows that $(X,\{f_\alpha\}_\alpha)$ forms a colimit in $C$.