I was interested in the definition of limit as universal arrow from $\Delta$ to $F$ given in Mac Lane's CWM book.
Let us begin with some notation: Let $C$ be a category, and $J$ be an index category, let $F: J \to C$ functor and let $\Delta : C \to C^J$ the diagonal functor (associate to each $c \in \text{Obj}(C)$ the constant functor $J \to F$ and to each map $f \in \text{Arr}(C)$ the natural transformation $\Delta f$ which has the same value $f$ at each object $i$ of $J$ (page 67 of CWM).
I cannot understand the one he gave for the limit (pag. 68):
a limit for a functor F is a universal arrow $\langle r,v \rangle$ from $\Delta$ to $F$.
But according to the definition of universal arrow (pag. 55), $\Delta$ should be an object of $C$ (because $F \colon J \to C$), but instead is an object of $(C^J)^C$. Can someone explain this discrepancy? Am I missing something?
The definition (as universal arrow) he gave for the colimit (pag. 67) was fairly clear, because the functor $F$ is indeed an object of $C^J$, hence the definition of universal arrow from $F$ (seen as an object) to $\Delta$ makes perfectly sense
The point is that the concept of universal arrow in Mac Lane's book is used in two directions: you have universal arrow from a functor to an object but also universal arrow from an object to a functor.
In the case of colimit an universal arrow is a natural transformation $u \colon F \to \Delta(r)$ where $F \in C^J$, $r \in C$ and $\Delta \colon C \to C^J$. For limit the situation is dual: an universal arrow is a natural transformation $u \colon \Delta(r) \to F$ where $F \in C^J$, $r \in C$ and (again) $\Delta \colon C \to C^J$.
The difference between these cases is that in the colimit case universality means that the arrow $u$ is initial between all the natural transformations $F \to \Delta(-)$ while in the limit case the arrow $u$ is terminal between all the natural transformation of type $\Delta(-) \to F$.
Hope this helps.