For any category $C$, with a terminal object $*$, denote by $C_*$ the coslice category $*/C$. There is an adjoint pair, denoted by $V_C\dashv U_C:C_*\to C$, where $U_C$ is the forgetful functor.
Let $C$, $D$ be two categories (wit terminal object), and let $F\dashv G:D\to C$ be an adjoint pair. There is an adjoint pair $F_*\dashv G_*:D_*\to C_*$ with the property that $GU_D=U_CG_*$; then there is a natural isomorphism $ F_*V_C\Rightarrow V_DF$, whose components are uniquely defined by the property of making this square commutative (for all $c$ in $C$):$\require{AMScd}$ $$\begin{CD} D_*(V_DFc,-)@>{\sim}>> C(c,GU_D-)\\ @VVV @| \\ D_*(F_*V_Cc,-)@>{\sim}>> C(c,U_CG_*-) \end{CD}$$
Now let $E$ be another category (with terminal object), and let $H\dashv K:E\to D$ be an adjoint pair. With these data, I get two natural isomorphisms $(HF)_*V_C\Rightarrow V_EHF$:
- By the observations before, a natural isomorphism is the unique one whose components make this square commutative: $$\begin{CD} E_*(V_EHFc,-)@>{\sim}>> C(c,GKU_E-)\\ @VVV @| \\ E_*((HF)_*V_Cc,-)@>{\sim}>> C(c,U_C(GK)_*-) \end{CD}$$
- I can obtain another natural isomorphism by horizontal composition, in a certain sense, and "correcting" $H_*F_*$ with $(HF)_*$: $$\begin{CD} E_*(V_EHFc,-)@>{\sim}>> C(c,GKU_E-)\\ @VVV @| \\ E_*(H_*V_DFc,-)@>{\sim}>> C(c,GU_DK_*-)\\ @VVV @| \\ E_*(H_*F_*V_Cc,-)@>{\sim}>> C(c,U_CG_*K_*-)\\ @VVV @| \\ E_*((HF)_*V_Cc,-)@>{\sim}>> C(c,U_C(GK)_*-) \end{CD}$$ The lower left vertical natural isomorphism is by definition the unique one making the lower square commutative. The upper two vertical equalities are due to this general fact (and its dual):
Let $L_0,L_1:A\to B$ and $R:B\to A$ three functors, and suppose that $L_0,L_1\dashv R$; then there is a (unique) natural isomorphism $\gamma:L_1\Rightarrow L_0$ whose components $\gamma_a$ make this square commutative. $$\begin{CD} B(L_0a,-)@>{\sim}>> A(a,R-)\\ @VV-\circ \gamma_a V @| \\ B(L_1a,-)@>{\sim}>> A(a,R-) \end{CD}$$ If $L'\dashv R':C\to B$ is another adjoint pair, $L'\gamma: L'L_1\Rightarrow L'L_0$ is the unique natural isomorphism whose components make the following square commutative. $$\begin{CD} C(L'L_0a,-)@>{\sim}>> A(a,RR'-)\\ @VV-\circ L\gamma_aV @| \\ C(L'L_1a,-)@>{\sim}>> A(a,RR'-) \end{CD}$$
If my arguments are correct, it follows that the natural isomorphisms of the point 1 and 2 coincide, as they make the same square commutative. My question is: is the fact in italics what Hovey means, when he says that $F_*V_C$ is naturally isomorphic to $V_DF$, and this correspondence is functorial (Proposition 1.3.5 in Model Categories)? I wouldn't be sure on how to turn rigourously the equality of isomorphisms 1 and 2 into an instance of functoriality, but this is the closest thing I could think of. What do you think? Thanks for your help