Good Internal Hom for Weak Complicial Sets?

So I am trying to learn a bit more about Dominic Verity's model of higher categories, namely weak complicial sets. The underlying object is a stratified simplicial set which satisfies a sort of inner horn filling condition. This model bears a resemblance to Joyal's quasi-categories, which is an exceptionally nice model of $(\infty,1)$-categories.

In Joyal's model there is a good inner hom, which is just the usual inner hom of simplicial sets. This inner hom is very well behaved and for well understood examples it computes the "correct" thing, the $(\infty,1)$-categories of "weak functors".

I am wondering if there is an analogous, easy description of the correct inner hom for weak complicial sets. What nice properties does it satisfy?

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I'm pretty sure that it's related to this tensor product: ncatlab.org/nlab/show/Verity-Gray+tensor+product –  Harry Gindi Jul 6 '11 at 22:04
Did you look at his 2006 weak complicial sets II paper. He gives a construction of an enrichment, but not directly. Another thought is to look at the T-complex stuff of Ronnie Brown and see if there is explicit description of the hom in that case. (I do not remember if one was given.) –  Tim Porter Nov 27 '11 at 9:24