Let $C$ be the category of finite simplicial complexes with simplicial maps. Then I want to define a functor $F : C \rightarrow C_n$, where $C_n$ is the subcategory of $C$ consisting of complexes up to dimension $n$, and where $F$ sends object $X$ to $\text{skel}^n(X)$ and sends morphism $f : X \rightarrow Y$ to $f|_{\text{skel}^n(X)} : \text{skel}^n(X) \rightarrow \text{skel}^n(Y)$. (If $\sigma \in \text{skel}^n(X)$, then necessarily $f(\sigma) \in \text{skel}^n(Y)$).
First of all, is this a functor? It clearly sends identity maps to identity maps, and it also seems to preserve compositions of morphisms, since simplicial maps are determined entirely by their underlying vertex maps, which are unchanged by $\text{skel}^n$.
Also, is there a name for this kind of construction? Contrast this with quotient categories, where we would identify morphisms. But in this case I am identifying objects (two objects are equal if they have the same $n$-skeleton), and morphisms are kind of unchanged (not quite, because their domains and codomains may change even though their underlying vertex maps do not).
Yes, this is a functor. The point is that you should think of simplicial complexes as being equipped with a canonical filtration by its skeleta, and more or less by definition, a simplicial map has to preserve the skeletal filtration. Taking the $n$-skeleton is just a forgetful functor that forgets the rest of the filtration. There's a simpler algebraic analogue involving the category whose objects are filtered vector spaces.