Let $\mathcal{C}$ be a cocomplete category, and let $F:\mathcal{C}\to\mathcal{C}$ be a functor with a natural transformation $i:1_\mathcal{C}\Rightarrow F$. If we have an object $c\in\mathcal{C}$, then we have a diagram $$c\xrightarrow{i_c}F(c)\xrightarrow{i_{F(c)}}F^2(c)\xrightarrow{i_{F^2(c)}}\cdots$$ of which we can take the colimit $\widehat{c}:=\operatorname{colim}_nF^n(c)$. Is it true that $\widehat{c}$ is a fixed point for $F$, i.e. that $i_{\widehat{c}}:\widehat{c}\to F(\widehat{c})$ is an isomorphism in $\mathcal{C}$?
This is not true in general, as pointed out by @Roland in the comments. What assumptions do we have to make on $F$ and $i$ in order for $\widehat{c}$ to be a fixed point?
(Of course, it is true if $i$ is a natural isomorphism, and it should also be true if $F$ commutes with colimits. I would like weaker conditions, if possible.)