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

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?

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.
• Thank you. I think I'm missing something big. $\mathbf{F}$ is supposed to be terminal in my example rather than initial. In $\mathrm{Hom}(\mathcal{C}, \mathcal{D})$, is the terminal object able to be a non-constant functor (if it exists)? May 20, 2021 at 19:58