# Existence of limits and colimits in a pointed category. [duplicate]

I am reading Mark Hovey's model category theory.

In the first chapter, on page $$4$$, we have a category $$\mathcal C$$ which has all small limits and colimits. He claims that the pointed category $${\mathcal C}_*$$ has arbitrary limits and colimits. He says:

Indeed, if $$F : I \to {\mathcal C}_∗$$ is a functor from a small category $$I$$ to $${\mathcal C}_∗$$, the limit of $$F$$ as a functor to $$\mathcal{C}$$ is naturally an element of $$\mathcal C_∗$$ and is the limit there.

I am not able to follow the argument. How did he think of $$F$$ as a functor to $$C$$? Did he forget the base point?

Yes, you can simply compose $$F$$ with the forgetful functor $$U: C_*\to C$$.
The argument is actually pretty simple : let $$*\to \lim(U\circ F)$$ be defined by the universal property of the limit applied to $$*\to U\circ F$$. This defines an object $$L\in C_*$$, and it's easy to check that the maps $$UL\to UF(i)$$ are pointed maps, indeed $$*\to \lim(U\circ F)\to UF(i)$$ is by definition the basepoint of $$F(i)$$, so we actually get a cone $$L\to F$$.