Adjoint pairs, triplets and quadruplets

Often we have adjoint pairs (A,B) (meaning A is left adjoint to B).
Sometimes we have adjoint triplets (A,B,C) (meaning A is left adjoint to B and B is left adjoint to C. No adjoint relation between A and C obviously, since they have the same source and target).

So the first question is: can we have quadruplets (A,B,C,D) ? (meaning that C is also left adjoint to D).

Second question:
It would be then possible that we have (A,D) besides (A,B). But I think not, because I think that it is not possible to have a functor A with two different right adjoints, B and D. Is this correct

Solutions Collecting From Web of "Adjoint pairs, triplets and quadruplets"

Actually, for every natural number $\ge 2$, there is a maximal string of adjoint functors of that length. Here’s one construction: let $\mathbf{n}$ denote the poset category $\{ 0 < \ldots < n – 1 \}$. I claim there are functors (i.e. order-preserving maps)
d_i & : \mathbf{n} \to \mathbf{n} + 1 & & 0 \le i \le n \newline
s_i & : \mathbf{n} + 1 \to \mathbf{n} & & 0 \le i \le n – 1
such that
$$d_n \dashv s_{n-1} \dashv d_{n-1} \dashv \cdots \dashv s_0 \dashv d_0$$
is a maximal string of adjoint functors. Moreover, if $\mathbf{C}$ is a category with a terminal object but no initial object, there are functors
\partial_i & : [\mathbf{n} + 1, \mathbf{C}] \to [\mathbf{n}, \mathbf{C}] & & 0 \le i \le n \newline
\sigma_i & : [\mathbf{n}, \mathbf{C}] \to [\mathbf{n} + 1, \mathbf{C}] & & 0 \le i \le n
such that
$$\partial_0 \dashv \sigma_0 \dashv \cdots \dashv \partial_n \dashv \sigma_n$$
is a maximal string of adjoint functors. (Here $[\mathbf{D}, \mathbf{C}]$ denotes the category of all functors $\mathbf{D} \to \mathbf{C}$.)

It is also not hard to find an infinite string of adjoint functors: for example,
$$\cdots \dashv \textrm{id} \dashv \textrm{id} \dashv \textrm{id} \dashv \cdots$$
is such a string, though a little degenerate.

As for your second question, recall that adjoints are unique up to unique natural isomorphism, i.e. if $F \dashv G$ and $F \dashv G’$, then there is a unique natural isomorphism $G \cong G’$ which interacts nicely with the unit and counit of the adjunction.