The diagonal functor $\Delta_C^J:C \to C^J$ and the constant functors $\Delta_C^J(c):J\to C$ definitions are a bit too generous and lead to contradictions when applied to $J=0$ (the initial category). Let's see why.
According to the definitions, for every object $c$ in $C$ $\Delta_C^0(c)$ is the unique functor $0\to C$. This functor is indeed vacuously constant.
This creates a contradiction between - for example - the following two statements in MacLane'CWM, 2nd ed, when you set $J=0$.
1- Page 90 exercise 8a :
If the category $J$ is connected, $\lim(\Delta_C^J(c))=c$
Indeed: $0$ is vacuously connected and $\Delta_C^0(c_1)=\Delta_C^0(c_2)$, yet their limits are different ($c_1$ and $c_2$), for any two non isomorphic $c_1$ and $c_2$ objects.
2- Page 71
"a limit of the empty functor to $C$ is the terminal object of $C$".
So the functor $0\to C$ does not always have a limit, and when it does, it does not follow from the formula $\lim(\Delta_C^J(c))=c$ above.
The other extreme case ($C=0$) also lead to questionable definitions.
So I think the definition of the diagonal functor should be limited to non empty $J$ and $C$.
Question: am I missing something? or do you agree?