I have been reading these great [notes](https://faculty.math.illinois.edu/%7Erezk/quasicats.pdf) by Charles Rezk, and one thing that has been bothering me is the join construction. To solve lifting problems in quasicategory theory we use the Leibniz construction, namely given a bifunctor $F: A \times B \to C$ that is bidivisible, namely sits in a two-variable adjunction $F \dashv \{ G, H \}$:
$$ C(F(a,b), c) \cong B(b, G(a,c)) \cong A(a, H(b,c))$$
there is an induced two-variable adjunction in the arrow categories $F' \dashv \{ G', H' \}$ with $F': A^I \times B^I \to C^I$. See Emily Riehl's Categorical Homotopy Theory Construction 11.1.7.


When $F: C \times C \to C$ has a two-variable adjunction such that $G \cong H$, we say it is closed. In $\mathsf{sSet}$ the bifunctor $- \times - : \mathsf{sSet} \times \mathsf{sSet} \to \mathsf{sSet}$ is bidivisible and closed, and the induced two-variable adjunction is the pushout product and pullback hom, which is used extensively to prove major theorems in quasicategory theory.


One can prove that the join construction $- \* -: \mathsf{sSet} \times \mathsf{sSet} \to \mathsf{sSet}$ is not bidivisible, because it doesn't preserve the initial object in each component. However, we can consider the functors $- \* X, X \* -: \mathsf{sSet} \to \mathsf{sSet}\_{X/}$. In this case they do have right adjoints, but they can't coalesce into a unique bifunctor since the codomain is changing for each $X$. In Rezk's notes, and in all other references that I could find, one still gets a more complicated induced adjunction and a resulting pushout join product and pullback slice hom, but we cannot prove their usual properties by the techniques developed for a bidivisible closed bifunctor, we have to do everything by hand.


Now there is an alternative construction where one takes $- \*' -: \mathsf{sSet} \times \mathsf{sSet} \to \mathsf{sSet}$ by giving the simplicial sets the augmentation $X\_{-1} = \pi\_0(X)$, and setting $(X \* Y)\_n = \coprod\_{i = -1}^n X\_i \times Y\_{n - 1 - i}$. This bifunctor is bidivisible, and when $X$ and $Y$ are connected, $X \* Y = X \*' Y$. So my question is: What are the objectwise adjoints to $- \*' -$? Can we use these to model the slice categories and develop the theory of (co)limits in quasicategories?



Join as a bifunctor
All of this is explained elegantly in Appendix D.2 in Riehl & Verity's Elements of $\infty$-category theory, and actually answers all of my questions. As Alexander Campbell says, the real important concept is the join as a bifunctor of augmented simplicial sets. Taking $\pi\_0$ or the trivial augmentation is not as important. We can use the functor category two-variable adjunction on augmented simplicial sets and then just forget the augmentations.


