How does a functor between index categories induce a canonical morphism between colimits? Let $\theta:I \rightarrow J$ be a functor between small categories. Let $C$ be a cocomplete category. Let $F:I \rightarrow C$ and $G:J \rightarrow C$ be two diagrams in $C$. Let $Colim(F)$ and $Colim(G)$  be colimits of  the diagram $F$ and $G$ respectively in $C$.
Given $\theta:I \rightarrow J$, does there always exists a natural choice of morphism $\theta_{*}:Colim(F)\rightarrow Colim(G)$ in $C$ ?
If not, then under what condition we can expect such a natural choice of morphism between colimits?
Thanks in advance.
 A: If $\theta$ maps over $C$ (in the sense that $F=G\circ\theta$), then there will be a canonical choice of morphism.
A cocone $(c,\alpha)$ for $G:J\to C$ (that is, an object $c\in C$ and morphisms $\alpha_j:G(j)\to c$ compatible with the morphisms in $J$) induces a cocone $(c,\alpha\circ\theta)$ for $F=G\circ\theta:I\to C$ since $\alpha_{\theta(i)}:F(i)=G(\theta(i))\to C$ respects the structure of $I$ by the functoriality of $\theta$. In particular, a colimit $\varinjlim G$ of $G$ induces a cocone of $F$.
Since a colimit $\varinjlim F$ of $F$ is a universal cocone for $F$, this means we must have a unique morphism $\theta_*:\varinjlim F\to\varinjlim G$ commuting with the cocone morphisms of both colimits.
However, if $\theta$ does not map over $C$, then there is no reason to expect there to be a canonical induced morphism of colimits.
In fact, there may not even be a morphism at all.
For a really simple example, let $I=J=*$ be the terminal category, and $C = \{0,1\}$ be the discrete category with two objects.
Take $F:I\to C$ to be the functor picking out $0\in C$ and $G:J\to C$ the functor picking out $1\in C$. We have a unique functor $\theta:I\to J$, but there is no morphism $\varinjlim F\to\varinjlim G$ as this would be a morphism $0\to1$ in $C$.
