Adjointness of the corresponding simplicial functors associated to an adjoint pair between categories

Question

Suppose you have an adjoint pair of functors $F:C\to D$ and $G:D\to C$. Let ${\mathrm{Simp}}C$ and ${\mathrm{Simp}}D$ be the category of simplicial objects in the categories $C$ and $D$ respectively. We have an associated pair of functors $\tilde{F}:{\mathrm{Simp}}C\to {\mathrm{Simp}}D$ and $\tilde{G}:{\mathrm{Simp}}D\to {\mathrm{Simp}}C$ between these simplicial categories. Is $\tilde{F}$ and $\tilde{G}$ still adjoint functors, at least up to homotopy?

Answer 1

Yes.

First something more general:

Let $A$ be a small category and let $F: C \leftrightarrow D : G$ be an adjoint pair of functors. Then there is an associated adjoint pair of functors $F^A : C^A \leftrightarrow D^A:G^A$ on the respective categories $C^A$ and $D^A$ of functors $A \to C$ and $A\to D$

To prove this, check:

1. Every functor $H: C \to D$ gives a functor $H^A: C^A \to D^A$ (simply by postcomposition of a functor $A \to C$ with $H:C \to D$).
2. Every natural transformation $\alpha: H \Rightarrow H'$ between functors $H,H': C \to D$ gives a natural transformation $\alpha^A : H^A \Rightarrow H'{}^A$.
3. Unit and counit of the adjunction $F: C \leftrightarrow D: G$ extend to unit and counit of an adjunction $F^A:C^A \leftrightarrow D^A:G^A$: verify the triangle identities (called counit-unit equations on Wikipedia).

Now observe that a simplicial object in $C$ is the same as a contravariant functor $\Delta \to C$, where $\Delta$ is the simplex category consisting of the finite ordinal numbers and order-preserving maps, so $\operatorname{Simp}{C} = C^{\Delta^{\rm op}}$.

Apply the above with $A = \Delta^{\rm op}$.