Limits in a functor category do what precisely to morphisms?

Question

So I know that if $\textbf{C}$ is a category and $\textbf{D}$ is a complete category then the functor category $\textbf{D}^\textbf{C}$ is complete and limits are "computed pointwise" in the sense that for each diagram $F : \mathcal{J} \to \textbf{D}^\textbf{C}$ and object $C$ of $\textbf{C}$ we have $(\lim_{j \in J} F_j)C = \lim_{j \in J} (F_jC)$. But my question is how are the morphisms computed e.g. for any $f : A \to B$ in $\mathcal{D}$ what is $(\lim_{j \in J} F_j)f$?

Answer 1

The same formula applies: $(\lim_j F_j)f = \lim_j F_jf$.

The precise way of stating this is to say that $\lim F : \textbf C → \textbf D$ is the composition $\lim{} ∘ \tilde F$, where $\tilde F : \textbf C → \textbf D^J$ is the functor $C ↦ F_{(-)}C$. Now given a morphism $f : A → B$, $\tilde F$ maps it to the natural transformation $(\tilde F_jf)_j : \tilde FA ⇒ \tilde FB$, and in general, given a natural transformation $α : G ⇒ H$ between two diagrams $J → \textbf D$, the functor $\lim$ associates to it the obvious induced morphism $\lim α : \lim G → \lim H$, ie. the one for which $π_j \lim α = α_j π_j$ holds. This last part, taking the limit of a natural transformation, is something that's often not mentioned (except for (co)products), but it's simple, and can often be useful.