Monotonicity of natural transformations between list and maybe functors Given two endofunctors $\mathit{List}, \mathit{Maybe}: \mathit{Set} \rightarrow \mathit{Set}$ defined as in Haskell with an ordering that preserves the functorial structure:
$$a_1 \ldots a_n \leq_{\overline{\mathit{List}}(A)} a_1' \ldots a_n' \text{ iff } a_1 \leq a_1' \ldots a_n \leq a_n'$$
$$\mathit{Nothing} \leq_{\overline{\mathit{Maybe}}(B)} \mathit{Nothing}$$
$$\mathit{Just}(b) \leq_{\overline{\mathit{Maybe}}(B)} \mathit{Just}(b') \text{ iff } b \leq_B b'$$
How can I prove that natural transformations $\eta: \mathit{List} \rightarrow \mathit{Maybe}$ are monotone?

 A: First let's classify the possible natural transformations $\eta: \text{List} \to F$ for any endofunctor $F$ on $\text{Set}$. We have $\text{List}(X) = \sum_{n \geq 0} X^n$, so $\text{List}$ is a coproduct of representable functors $\hom(n, -): X \mapsto X^n$. The set of natural transformations is therefore 
$$\text{Set}^\text{Set}(\text{List}, F) \cong \text{Set}^\text{Set}(\sum_{n \geq 0} \hom(n, -), F) \cong \prod_{n \geq 0} \text{Set}^\text{Set}(\hom(n,-), F) \cong \prod_{n \geq 0} F(n)$$ 
where the last isomorphism is by the Yoneda lemma. For $F = \text{Maybe}$ in particular, a transformation $\eta: \text{List} \to \text{Maybe}$ is specified by a sequence of integers $\mathbf{n} = (n_0, n_1, \ldots)$ where $0 \leq n_k \leq k$ for each $k \geq 0$. It is defined by the rule 
$$\eta_\mathbf{n}(x_1\ldots x_k) := x_{n_k}$$ 
if $1 \leq n_k \leq k$, and $\eta_\mathbf{n}(x_1\ldots x_k) = \mathit{Nothing}$ if $n_k = 0$. 
So now I think we just follow our noses. If $x_1\ldots x_k \leq y_1\ldots y_k$ in $\text{List}(X)$, then for each $\mathbf{n} = (n_k)_{k \geq 0}$ we have (in case $1 \leq n_k \leq k$) 
$$\eta_\mathbf{n}(x_1\ldots x_k) = x_{n_k} \leq y_{n_k} = \eta_\mathbf{n}(y_1 \ldots y_k)$$ 
where the displayed $\leq$ follows from the definition of $\leq$ on $\text{List}(X)$, with a similar inequality in case $n_k = 0$. So indeed the transformation $\eta_\mathbf{n}$ is monotone. 
