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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?

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Yes, you can simply compose $F$ with the forgetful functor $U: C_*\to C$.

The claim is that this forgetful functor creates limits.

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 $L\to F$.

Checking that it's a limit is straightforward.

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