Let $C$ be a category with finite limits; that is, for any finite category $D$ and functor $F:D\to C$ the category $\mathrm{Cone} F$ of cones over $F$ is inhabited and has terminal objects (we could turn the morphisms around and call the limit an initial obect, but never mind).Fix such a
That is, define $D$ $ (A,\phi): \mathrm{Cone} F \iff \phi:Nat(\Delta_A, F) $$ where $\Delta_A(d) = A$ and $F$. There \Delta_A(\delta:d\to d')= 1_A$; and for $(A,\phi),(B,\psi):\mathrm{Cone} F$ define $$ \mathrm{Cone}F((A,\phi),(B,\psi)) = \{ f:A\to B \mid \phi_d = \psi_d\circ f\} .$$ A limit for $F$ is a terminal object $(H,\eta)$ in $\mathrm{Cone}F$.
Now there is an obvious forgetful functor $F^\perp: \mathrm{Cone} F \to C$ . with $F^\perp(A,\phi)=A$. More: for any limit $\eta = (\eta_d:H\to F(d))_ d $(H,\eta)$ there is a co-cone $\chi = (\chi_\phi:\Phi\to H)_ \phi$, (H,\chi)$ with $\chi_\phi:F^\perp(\phi)\to H$, the unique one showing that $\eta$ (H,\eta)$ IS a limit. For any co-cone $c:F^\perp \to A$ (A,c)$ under $F^\perp$, there is by hypothesis a morphism $c_\eta:H\to A$, again by hypothesis making everything commute that should; thus $\chi$ ( $\eta$ (or $H$)) is versal for cocones under $F^\perp$.
I'd like to eventually conclude that $H$ is universal --- that $c_\eta$ is the only thing making everything commute where it should, so that $\chi$ makes $H$ a colimit for $F^\perp$, but from the diagrams, I'm afraid it probably doesn't.
Are there supplementary assumptions that'll make everything work out nicely? Does it work out nicely and I just don't see it?
Or maybe I'm wrong about something!? That'd be OK, too.
