The object map $\mathrm{lim}:\sf C^J\to C$ is known to extend to a functor; does this fact is already proven by the fact that the limit of a diagram of functors is computed objectwise?
If $F:\sf I\to C^J$ is a diagram, we can view it as a diagram of functors $\bar F:\sf J\to C^I$, so that $\operatorname{lim}\bar F$ is a functor $\sf C^I$. Since $(\operatorname{lim}\bar F)(i)=\operatorname{lim}(F(i))$ for every $i\in \operatorname {ob} I$, setting $\mathsf {I}=2$, the category with two object and one non-identity arrow, we obtain an extension of $\operatorname{lim}:\sf C^J\to C$ to the arrows; setting $\mathsf{I}=3$, the category with three objects and three arrows forming a commutative triangle, we see that it is functorial, and similarly for the identity using $1$. I feel like I'm missing something, because maybe we can get a global description as a functor $\sf C^J\to C$, instead of noticing that it basically preserves commutative diagrams, to prove its functoriality. I hope that I made my doubt clear, thanks.