Let $I$ be a small category, $A$ a category and $D:I\rightarrow A$ a functor. A cone on $D$ is an object $a\in A$ together with a family $(a\xrightarrow{f_i}D(i))_{i\in I}$ of maps in $A$ such that for all maps $i\xrightarrow{u}j$ in $I$, $D(u)\circ f_i=f_j$.(and then we define a limit of $D$)
I was wondering if such cones exist.
Not always.
Consider the category $\mathcal{C}$ with two objects $X,Y$ and two distinct arrows $f,g:X\rightarrow Y$. Consider the identity functor $Id_\mathcal{C}:\mathcal{C}\rightarrow \mathcal{C}$. There is not a cone above this functor:
The object $Y$ can not be a cone because there are not arrows $Y\rightarrow X$;
The object $X$ can not be a cone because the arrow $X\rightarrow X$ is forced to be the identity but the arrow $X\rightarrow Y$ has to be $f$ and $g$ at the same time for the commutativity of the diagram; this is a contraddiction since $f\neq g$.