Question: I want to define terminal objects from the categorical definition of limit, but I cannot. What is my mistake in the following argument?
Limit: Given a functor $F: \mathcal{D} \longrightarrow \mathcal{C}$, a limit of $F$ is a cone $L$ such that for every $M$, these following sets are naturally isomorphic $$\mathcal{C}(M, L) \cong \textbf{Nat}(\Delta_M, F)$$ ($\Delta_M$ is the constant functor and Nat is the set of natural transformation between these two functors $\Delta_M, F$)
Recovery of terminal object: Let the category $\mathcal{D}=\emptyset$ be the empty category, and $F=\Delta_{L}$. Then, for every $M$, $$\mathcal{C}(M, L) \cong \textbf{Nat}(\Delta_M, \Delta_{L})$$ $\Delta_L$ assignes $L$ to the empty category. So this isomorphy is trivial because both sets are the same. But it does not say $\mathcal{C}(M, L) $ has only one element, or $M \longrightarrow L$ is unique.
Where is my mistake?
Notice: The question here is regarding the categorical definition of limit and I am not asking for an explanation, I am looking for my mistake in my argument. (Response to a possible duplicate of other questions! )
It's not clear what you mean by "$\Delta_L$ assigns $L$ to the empty category".
For any set $S$, there is a unique function $\varnothing \to S$. Using this function where appropriate, you can see:
Consequently, $\textbf{Nat}(\Delta_M, \Delta_{L})$ is the one point set $\{ \epsilon \}$.