Recall that extra degeneracies for an augmented simplicial set $X$ are maps $s_0\colon X_n \to X_{n+1}$ for $n=-1,0,1,2,\ldots$ which satisfy the usual simplicial identities with respect to the existing $d_i$, $s_i$. This definition clearly works for simplicial objects in pretty much an arbitrary category.

A simplicial set with extra degeneracies and $X_{-1} = \ast$ (call these, in abuse of nomenclature, reduced) is contractible. In fact, given enough structure on an ambient category $C$, one can sensibly talk about homotopy of simplicial objects in $C$ (for example, one can say when $sC$ is a category with cofibrant objects, and has a notion of homotopy of maps).

Tim Porter points out here an observation (probably due originally to Lawvere) that if we define $\Delta_{last}$ to be the subcategory of $\Delta$ of non-empty finite ordinals whose morphisms preserve the last element in each ordinal, then $C$-valued presheaves on $\Delta_{last}$ are equivalent to augmented simplicial objects with extra degeneracies. One can then consider, if $C$ has a terminal object, the subcategory of 'reduced' presheaves. Presumably we can consider these as functors preserving something (terminal object? Please excuse my laziness for not checking this, it's not crucial to the question).

So my question is this: is it reasonable to think of reduced presheaves on $\Delta_{last}$ as being contractible for any $C$ with terminal object? Certainly, ignoring size issues, we can think of such things as being contractible after we embed them in the category of simplicial sets in $Pre(C)$, if not some smaller (co)completion category.

Secondarily, can I get away with saying a reduced presheaf on $\Delta_{last}$ "is a contractible simplicial object?" If I define such as thing for $C$ with insufficient structure to support homotopies as a reduced presheaf on $\Delta_{last}$ then it all becomes a bit tautological.

EDIT: removed cruft from this question.

Recall that extra degeneracies for an augmented simplicial set $X$ are maps $s_0\colon X_n \to X_{n+1}$ for $n=-1,0,1,2,\ldots$ which satisfy the usual simplicial identities with respect to the existing $d_i$, $s_i$. This definition clearly works for simplicial objects in pretty much an arbitrary category.

A simplicial set, augmented by $X_{-1} = \ast$, with extra degeneracies is contractible. In fact, given enough structure on an ambient category $C$, one can sensibly talk about homotopy of simplicial objects in $C$ (for example, one can say when $sC$ is a category with cofibrant objects, and has a notion of homotopy of maps). [EDIT: By this I mean there is some sort of model structure around relative to which we can talk about homotopy]

So my question is this: is it reasonable to think of simplicial objects in $C$ with extra degeneracies as being contractible for any category $C$ with terminal object? Certainly, ignoring size issues, we can think of such things as being contractible after we embed them in the category of simplicial sets in $Pre(C)$, if not some smaller (co)completion category.

Secondarily, can I get away with saying a simplicial object with extra degeneracies "is a contractible simplicial object?"