Say you have a bunch of posets consisting of partial orders over the set of, let's just say, letters in the English language (a, b, c, etc). Say you collect a bunch of these posets as objects of a discrete category, let's name it $\textbf{Conf}$ for "confused," because I am. I'm wondering if it is possible to talk about constructing a particular poset over the set of English letters from the objects in this category. Suppose for example that you have 26 posets in the category labeled A, B, C, etc., and luckily $a$ is the "greatest" object in $A$, $b$ is the "second greatest" object in $B$, etc. It seems like it should be perfectly logical to say that you can draw a functor from this category to an individual poset that consists of $ z \leq ... \leq b \leq a$. It would map $A$ to $a$ and so on, and the empty set of morphisms from $B$ to $A$ to the singleton set of morphisms from $b$ to $a$ and so on. In fact, I don't think you even need to say anything about $a$ being greatest in $A$, $b$ being second greatest in $B$, etc., nor about having 26 posets. You can just map however you like, really.
But say you have a different rule. Say that $a$ is the terminal object for all objects of $\textbf{Conf}$. Is it possible to have a functor that says "if an element is a terminal object for all the posets in $\textbf{Conf}$, then make it the terminal element in the poset that is the target of the functor from $\textbf{Conf}$"? On the one hand, doing so seems extremely intuitive. On the other hand, I don't understand how to do so. The objects of $\textbf{Conf}$ "are" posets, but only in the same way that, say, the elements of the set of mountain ranges "are" mountain ranges. It is not like the set of mountain ranges would be incredibly heavy to lift. Elements of sets don't have terminal objects, that's something a category itself has, so how can the posets in $\textbf{Conf}$ have terminal objects, and therefore, how can such a rule be constructed? I feel like this should be incredibly obvious, but I can't figure it out. Basically, I feel like collecting the posets into $\textbf{Conf}$ "deletes" the morphisms within each individual poset, and so I don't understand conceptually how to refer back to them.
I thought that I had thought of a way to do it, which is to stick all the morphisms of each of the posets in $\textbf{Conf}$ into a single category with the letters of the English alphabet as objects and a morphism from $a$ to $b$ for each instance that a poset in $\textbf{Conf}$ has a morphism from $a$ to $b$, and a morphism from $b$ to $a$ for each instance that a poset in $\textbf{Conf}$ has a morphism from $b$ to $a$. So this category is no longer a poset (I don't know what it's called anymore), but it doesn't "delete" the morphisms. Then if $a$ was terminal for every poset in $\textbf{Conf}$, it would still be terminal in this new category. But I don't believe that you can in general construct a poset from this new category since if it's true in this new category that $a \to b$ and $b \to a$, then you cannot establish a functor from this new category to a poset since you will either fail to preserve composition or violate antisymmetry of the target poset.
So...I guess my question is really just: how do you get a poset out of a collection of posets, and how exactly should you collect them in order to do it? And relatedly, what am I doing wrong? Here's some extra context.