When does the category of covariant functors from $\mathcal{C}$ to $\mathrm{Set}$ have an initial object?

Question

I'm trying to understand functors going to $\mathrm{Set}$ since they appear in discussions of Grothendieck topologies, and the Yoneda Lemma. I have a basic question about the structure of the collection of these functors coming from a fixed category $\mathcal{C}$.

Let $\mathcal{C}$ be a category.

Let $\mathcal{C}_*$ nonstandardly denote the category of covariant functors from $\mathcal{C}$ to $\mathrm{Set}$, where the arrows in $\mathcal{C}_*$ are natural transformations as usual.

Let $\mathbf{F} : \mathcal{C} \to \mathrm{Set}$ be the functor that sends all objects to $\{\varnothing\}$ and all arrows to $\{(\varnothing, \varnothing)\}$. Since $\mathbf{F}$ collapses all arrows and all objects, I'm pretty sure that any other functor will have a unique natural transformation sending it to $\mathbf{F}$. So, $\mathbf{F}$ is a terminal object (up to unique isomorphism). I think. This makes intuitive sense if I think about a loose analogy to the category of Rings without identity, where the initial object is $\mathbb{Z}$ in which I don't identify anything unless I have to and the final object is $\{0\}$ where I identify everything.

Under what circumstances does $\mathcal{C}_*$ have an initial object?

Answer 1

The category $\mathrm{Hom}(\mathcal{C},\mathbf{Set})$ of functors $\mathcal{C} \to \mathbf{Set}$ always has an initial object, namely the constant functor whose value is some initial object of $\mathbf{Set}$. More generally, if $\mathcal{C}$ is any category and $\mathcal{D}$ has an initial object $0$, then $\mathrm{Hom}(\mathcal{C},\mathcal{D})$ has an initial object, namely the constant functor $\Delta(0)$. (Even more general statements can be made about the existence of colimits.) The proof is straight forward: A morphism $\Delta(0) \to F$ is a family of morphisms $0 \to F(c)$ (where $c \in \mathcal{C}$) such that for all $c \to c'$ the naturality square commutes. But $0 \to F(c)$ is uniquely determined since $0$ is initial, and the square commutes automatically since $0$ is initial.

It follows formally that $\mathrm{Hom}(\mathcal{C},\mathcal{D})$ has a terminal object as soon as $\mathcal{D}$ has one. For example, you can deduce this from $\mathrm{Hom}(\mathcal{C},\mathcal{D})^{\mathrm{op}} \cong \mathrm{Hom}(\mathcal{C}^{\mathrm{op}},\mathcal{D}^{\mathrm{op}})$. But it is again just the constant functor for a terminal object of $\mathcal{D}$.

For $\mathcal{D}=\mathbf{Set}$, this gives basically the terminal functor which you describe, but I would suggest to not write functions as sets, this is a "type error" built into ZFC.