# Limits and colimits are computed pointwise in functor categories

There is a slogan that limits and colimits are computed pointwise in functor categories. I was trying to show it for pullbacks, but I couldn’t. Then, how can we prove it at least for pullbacks?

If $$D^C$$ denotes the category of functors $$C\to D$$ and we have a diagram $$a:F_1\to F_3\leftarrow F_2:b$$ in $$D^C$$, then we define the pullback $$P:C\to D$$ by letting $$P(c)$$ be the pullback of $$F_1(c)\to F_3(c)\leftarrow F_2(c)$$ and $$P(f:c\to c’)$$ the induced morphism between the pullbacks. (This uses functoriality of pullbacks and naturality of the morphisms between the $$F_i$$.)
Now given a functor $$G:C\to D$$ with natural transformations $$\alpha:G\to F_1,\beta:G\to F_2$$ such that $$a\circ \alpha=b\circ \beta$$, we have for every $$c$$ that $$a_c\circ \alpha_c=b_c\circ \beta_c$$, which uniquely determines a morphism $$G(c)\to P(c)$$. The fact that these morphisms assemble into a natural transformation $$G\to P$$ is all that remains to be shown; it follows again from various naturality squares and I leave it to you to finish up.