The n-Category Café

First suppose $F$ is a parametric right adjoint, with left adjoint $L : D/F1\to C$ of $F_1$. Then given $y\in D$, let $I = D(y, F1)$ (which is a set since $D$ is locally small). Then each $i\in I$ is a morphism $i:y\to F1$, i.e. an object of $D/F1$, so we have $L i \in C$ and the unit $\eta_{y,i}: y \to F L i$ a morphism in $D/F1$. Now suppose given $g:y\to F z$. Then the composite $y \to F z \to F 1$ is an element $i\in I$, making $g$ into a morphism of $D/F1$. Thus, it corresponds to a unique morphism $L i \to z$, giving the unique factorization in the definition of multi-adjoint.

Conversely, suppose $F$ has a left multi-adjoint, and let $k:y\to F 1$ be an object of $D/F1$. Let $\{ \eta_{y,i}:y \to F x_i\}_{i\in I}$ be given as in the definition of multi-adjoint; then there is a unique $i\in I$ and a factorization of $k$ through $\eta_{y,i}$ (necessarily by the unique morphism $x_i \to 1$). Define $L k = x_i$. Then for any $z\in C$, we know that any map $g:y\to F z$ factors uniquely through $\eta_{y,j}$ for some $j\in I$. But it follows that the composite $y \to F z \to F 1$ must factor through the same $\eta_{y,j}$; so if $g$ is a morphism in $D/F1$ then we must have $j=i$, so that $g$ factors uniquely through $L k$. Thus, $L$ is a left adjoint to $F_1$. This completes the proof.

In fact, this argument is essentially already in the literature. In Familial 2-functors and parametric right adjoints, Weber proved (Prop. 2.6) that a functor whose domain has a terminal object is a parametric right adjoint if and only if every morphism $f:y\to F z$ factors as $y \xrightarrow{g} F x \xrightarrow{F h} F z$ where $g$ is “$F$-generic”. The definition of $F$-generic involves a sort of restricted lifting property, but it’s easy to see that the units $\eta_{y,i}$ in a multi-adjoint are $F$-generic, and that if $F$-generic factorizations exist then a representative set of their isomorphism classes forms a multi-adjoint (up to size questions).

However, I haven’t seen this stated before using the terminology “multi-adjoint”. In particular, it’s interesting that multi-adjoints can be thought of as a generalization of parametric adjoints to categories lacking a terminal object.