The definition of a simplicial set is given as a contravariant functor $\textbf{F}:\Delta\to\textbf{Set}$ where $\Delta$ is the category of finite ordinals with order preserving functions as morphisms. The way I understand this definition and how it is a good model for topological simplicial complexes is that the image of each $[n]$ represents the n-dimensional simplex and the images of the coface and codegeneracy maps tell you how the simplexes fit together.
This all makes sense to me (though please tell me if there's some conceptual error here) but what I don't understand is why we use a contravariant functor rather than a covariant one. It seems to me that if we just swapped the names of coface and codegeneracy maps and changed it to a covariant functor we would get the same result. I'd love to either get confirmation that this is the case and the choice is purely historical or get some intuition on how these two definitions differ and why contravariant is a better choice.
Thanks for taking the time to read my question
First, let's look at the Simplex Category $\Delta$. We can see that $\Delta$, rather than $\Delta^{op}$, is a natural category to look at, both because it has a nice combinatorial description (the objects are finite total orders and the morphisms are order preserving maps) and, more importantly for us, because it has a topological visualisation which goes as follows:
Therefore, we expect that our functors will have source $\Delta$ and would prefer to make functors contravariant rather than switch to having $\Delta^{op}$ as the source.
Now lets look at what's going on with a simplicial set $X$ and see how we can construct it as a functor. The set $X_n$ is the set of all instances of $n$-simplices in the simplicial set. Now, for any $n$-simplex we can pick out each of its faces: for each choice of face we get a function $X_n \rightarrow X_{n-1}$. But this function was constructed from the face map $[n-1] \rightarrow [n]$: it is a function sending each $n$-simplex to the $n-1$-simplex that was included into it by that. So, face maps should be sent to functions going the other way round.
Similarly, for each $n$-simplex we can construct a degenerate instance of an $n+1$-simplex occupying the same space. We can view this constructed $n+1$-simplex as the map that first projects and $n+1$-simplex onto a face using a degeneracy map, then includes the original $n$-simplex into the simplicial set. Thus the degeneracy maps $[n+1] \rightarrow [n]$ are turned into functions $X_n \rightarrow X_{n+1}$.
Combining this, we see that $X$ is a covariant functor $\Delta^{op} \rightarrow \underline{Set}$. Since $\Delta$ is a more natural category to work with and can be viewed as a category consisting precisely of the simplices as topological spaces and continuous maps, we think of this as being a contravariant functor $\Delta \rightarrow \underline{Set}$.