Proving that the category of forests $\mathcal{F}$ is isomorphic to $\text {Set} ^{\mathcal{C}^{op}}$

Question

I just started studying category theory. So far I learned the basic definitions of Categories, Functors, Natural transformations, and saw some examples. I want to show that the category of forests $\mathcal{F}$ is isomorphic to $\text{Set} ^{\mathcal{C}^{op}}$, with $\mathcal{C}$ a small category.

The definition that I'm using of forest is the following:

A forest is a poset $(T,<)$ such that $\forall x\in T, T_x:=\{ y\in T \vert y<x \}$ is a finite totally ordered set.

My attempt:

[I made two silly mistakes below, pointed out in the comments - lost track of what $H(T)$ was and was thinking wrongly about forests. After realizing this with the help of the comments, the question vanished]

I want to find functors $H:\mathcal{F}\rightarrow \text{Set}^{\mathcal{C}^{op}}$ and $G:\text{Set}^{\mathcal{C}^{op}}\rightarrow \mathcal{F}$ such that their compositions $GF$ and $FG$ are identities. In particular, I want $H(T)$ to be a functor $\mathcal{C}^{op}\rightarrow \text{Set}$. Take $\mathcal{C}=\mathbb{N}$ (viewed as a poset with the usual order), which is small. Also, suppose $T$ is countable. To construct $H(T)$ we do the following: we number the elements of $T$ by level of the forest, as illustrated in the picture.

Now, call $R\subseteq T\times T$ the order in $T$. For each pair in $R$, extract the corresponding pair of $\mathbb{N}\times \mathbb{N}$. The set of all such pairs is $H(T)$. In the example of the picture, $H(T) = \{ (0,2),(0,3),(0,5),(2,5),(3,5),(1,3),(1,4),(1,6),(4,6) \}$

$G$ is now defined as reversing this process. This gives us a problem as described below.

We still have to define how $H$ and $G$ act on arrows. Given a forest morphism $f:T\rightarrow T'$ (which preserves order and levels of a forest), we assign a map $H(f)$ between the sets $H(T)$ and $H(T')$ that assigns to each pair $(n(a),n(b))\in H(T)$ the pair $(n(f(a)), n(f(b)))$ where $n$ is the map assigning natural numbers to elements of a forest as (partially) described above. Again, $G$ is very naturally defined as reversing this.

Problems:

1. $G$ should take a set of pairs of natural numbers and give us a forest. But there is an ambiguity when constructing the forest: how to construct it in a way that reverses what $F$ does? e.g. when $F$ acts as described above, it may assign the number $4$ to an element to which $G$ assigned the number $3$, if these are in the same level of the forest (as in the picture above, for example). This problem can be restated as follows: my function $n$ above was not well defined, because I don't know how to assign numbers to the elements of a forest in a unique way.
2. I had to assume that $T$ was countable, which seems to be false in general. So maybe I could try to do the same but using $\mathbb{R}$ intead of $\mathbb{N}$, but I am not sure exactly how to do it rigorously.

I hope this is clear enough.

Answer 1

How about $\mathcal C$ to be the set of non-negative integers considered as an ordered set?

The image of $0$ is the set of roots, and the map $n\mapsto {n-1}$ induces the map of nodes of distance $n$ from the root to their parents.