Let $[\mathcal{C},\mathcal{D}]$ denote the category of functors $\mathcal{C}\to\mathcal{D}.$
My notes say that if $\mathcal{D}$ has limits of shape $I$, then the composite $$[I, [\mathcal{C},\mathcal{D}]] \cong [\mathcal{C},[I,\mathcal{D}]] \to [\mathcal{C},\mathcal{D}]$$ is the same as the limit functor, where the right arrow comes from the limit functor $[I,\mathcal{D}] \to \mathcal{D}.$
I understand this statement, and the proof, but I'm having a hard time grasping why this means we can compute limits pointwise (or even what this means in the first place). Does it mean that:
Suppose $F$ is a functor $I \to [\mathcal{C},\mathcal{D}].$ We want to calculate $\lim_I{F}.$ First, change $F$ to a functor $G \colon \mathcal{C} \to [I, \mathcal{D}]$ by defining $G(x)(i) = F(i)(x)$ for $i \in I$ and $x\in \mathcal{C}.$ Then compose this with the functor $\lim_I \colon [I,\mathcal{D}] \to \mathcal{D}.$ So when we say "pointwise", do we mean that $$\lim_I{F}(x) = \lim_I(Gx)?$$