Let $D:I\to[\mathscr A,\mathscr C]$ be a diagram, and $ev_A:[\mathscr A,\mathscr C]\to \mathscr C$ the evaluation-at-$A$ functor. Let $L:\mathscr A\to \mathscr C$ be the functor such that $L(A)$ is the vertex of the limit cone of $ev_A\circ D=D(-)(A)$. There's a cone $(p_i:L\to D(I))_{i\in I}$ on $D$ in $[\mathscr A,\mathscr C]$, where $p_i$ is the natural transformation whose component at $A$, $p_{i,A}$, is the projection from the limit $L(A)$ to $D(i)(A)$.
I'm trying to prove that this cone $(p_i)_{i\in I}$ is a limit cone. Given another cone $(f_i:K\to D(i))_{i\in I}$, I can define $\kappa:K\to L$ as follows: the $A$th component of the natural transformation $\kappa$ is the unique map from $ev_A(K)=K(A)$ to the limit $L(A)$.
But I'm having difficulties with showing that $\kappa$ forms a natural transformation. I need to show that for any $s:A\to A'$, one has $\kappa_{A'}\circ K(s)=L(s) \circ \kappa_A$. How to do that? I looked at my old notes, and there I followed both arrows by $p_{i,A'}$ and claimed that the triple compositions are equal (and after that deduced that the double compositions are equal). But I don't see now why the triple compositions are equal.
Addition: $\mathscr C$ is assumed to have limits of shape $I$ (or at least the diagram $D(-)(A)$ is assumed to have a limit). Also I guess $\mathscr A$ and $I$ are assumed to be small and $\mathscr C$ is assumed to be locally small.