$\Delta[n]$ to define the ‘hom’ and the composition, so is very much like (i).

In general any category of simplicial objects in a ‘nice enough’ category has a simplicial enrichment, although the general argument that gives the construction does not always make the structure as transparent as it might be.

There is an evident notion of $\mathcal{S}$-enriched functor, so we get a category of ‘small’ $\mathcal{S}$-categories, denoted $\mathcal{S}\!-\!Cat$. Of course, none of the above examples are ‘small’. (With regard to ‘smallness’, although sometimes a smallness condition is essential, one can often ignore questions of smallness and, for instance, consider simplicial ‘sets’ where actually the collections of simplices are not truly ‘sets’ (depending on your choice of methods for handling such foundational questions).)

From simplicial resolutions to $\mathcal{S}$-cats.
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The forgetful functor $U : Cat \to DGrph_0$ has a left adjoint, $F$. Here $DGrph_0$ denotes the category of directed graphs with ‘identity loops’, so $U$ forgets just the composition within each small category but remembers that certain loops are special ‘identity loops’. The free category functor here takes, between any two objects, all strings of composable *non-identity* arrows that start at the first object and end at the second. One can 