The join of two categories $C$ and $C'$ is obtained from the disjoint union of $C$ with $C'$ by throwing in a unique morphism from every object of $C$ to every object of $C'$.
The join of categories $C$ and $C'$ is the category with
objects $Obj(C \star C') := Obj(C) \amalg Obj(C')$;
morphisms given by
The join of categories $C,C'$ can also be described to be the cograph of the unique profunctor $W\colon C ⇸ \; C'$ sending all objects $(c,c')$ to the terminal set (the definition speaks for itself).
Consider the inclusion of the boundary of the standard 1-simplex, $i\colon \{0,1\}\to [1]$ as a functor between the discrete category with two elements and the walking arrow $I=\{0 \leq 1\}$. It induces a functor
which admits a right adjoint. This right adjoint is precisely the bifunctor $\star\colon \Cat \times \Cat \to \Cat / I$, once we noticed that the category $C\star C'$ comes naturally equipped with an arrow $C\star C'\to I=1\star 1$ induced by (bi)functoriality of $\star$, starting from the canonical arrows $C\to 1, C'\to 1$ to the terminal category.
It is quite clear that $i^*$ is defined by sending $C\to I$ to the pair of categories $i^\leftarrow(0)=C_0, C_1=i^\leftarrow(1)$. The bijection
is now rather obvious, since any functor $i^* \Big( \array{ C\\ \downarrow \\ I } \Big) \to (A,B)$ determines a functor $C\to A\star B$ and viceversa.
See p. 42 of
See also Ch. 3 of
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