THC situations and the projection formula I'm trying to find the real nature of the so-called "projection arrow" for $f^* \dashv f_* $:
$$
f_*(A\otimes f^*B)\leftarrow f_*A\otimes B
$$
which can be found in any monoidal category $(\mathbf V,\otimes)$, and sometimes is an isomorphism (for example, when you look at categories of sheaves and there are suitable hypoteses of flatness over $A,B$).
My problem is that the machinery of "THC-situations" proposed in Gray's paper "Closed categories, lax limits and homotopy limits" seems to apply... backwards, and I'm not able to reverse it.
Let me recall a couple of things:


*

*A THC situation consists of three categories $A_1,A_2,A_3$ and functors $T\colon A_1\times A_2\to A_3$, $H\colon A_2^o\times A_3\to A_1$, $C\colon A_1^o\times A_3\to A_2$ such that
$$
A_3(T(a_1,a_2),a_3)\cong A_1(a_1,H(a_2,a_3))\cong A_2(a_2, C(a_1,a_3))
$$
$T,H,C$ have respectively to be regarded as generalized tensor, hom and cotensor operations (a paradigmatic example of a THC situation is that of a monoidal closed category, where $T=-\otimes-, HC=[-,-]$). One can denote a THC situation as $\mathfrak t = \lbrace A_1,A_2,A_3,\begin{pmatrix}T\\ H\\ C\end{pmatrix} \rbrace$.

*One is now interested in linking two THC situations $\mathfrak t, \mathfrak t'$ via adjoint pairs. In what follows $\check G_i \dashv G_i\dashv \hat G_i$ is a string of adjoint functors where $G_i\colon A_i\to A_i'$, for $i=1,2,3$. Now it's easy to prove, for example, that
$$
G_3T(\check G_1-,\check G_2-),\;\; G_1H(\check G_2-,\hat G_3-),\;\ G_2C(\check G_1-,\hat G_3-)
$$
is a THC situation.

*It's also easy to prove that $G_1H(a_2,a_3)\cong H'(G_2a_2,G_3 a_3)$ if and only if $T(\check G_1 a_1',a_2)\cong \check G_3T'(a_1',G_2a_2)$, by simple adjoint calculus and the Yoneda Lemma: on the one hand
$[a_1', G_1H(a_2,a_3)]  \cong [\check G_1 a_1', H(a_2,a_3)] \cong [T(\check G_1a_1', a_2), a_3]$, and on the other hand $[a_1',H'(G_2a_2, G_3a_3)] \cong [T'(a_1',G_2a_2),G_3a_3] \cong [\check G_3T'(a_1',G_2a_2),a_3]$.


This should have now the following pleasant corollary: if I am given two monoidal closed categories $(\mathbf V,\otimes, [-,=])$, $(\mathbf V',\otimes', [-,=]')$ and a couple of adjoint functors $F\colon \mathbf V\dashv \mathbf V'\colon G$, then $G[V,W]\cong [GV,GW]'$ iff $FV'\otimes W\cong F(V'\otimes' GW)$, which is exactly what I need... backwards, because what I need is that the right adjoint behaves like this:
$$
 GV'\otimes W\cong G(V'\otimes' FW)
$$
in order to justify the projection formula. Is there a similar TH(C) situation where I can get what I want?
 A: Here are some references that you may find helpful.


*

*The most common modern term for "THC-situation" is a two-variable adjunction.  They can indeed be composed with ordinary adjunctions as you suggest.  More generally they are the binary morphisms in a multicategory, and indeed a polycategory,  whose unary morphisms are ordinary adjunctions.

*The equivalence between a right adjoint being closed monoidal and a "Hopf/Frobenius" condition relating it to its left adjoint is also standard.  This is sometimes called "Frobenius reciprocity" and other times a Hopf adjunction.

*The "projection formula" map is indeed "backwards" from the Hopf adjunction map.  The only abstract study I know of that mentions it is Fausk-Hu-May.  Their equation (3.6) gives a general way to define such a morphism for any lax monoidal right adjoint, but they don't mention any "mate" property for it analogous to the Hopf-adjunction/closed-functor duality, and in general it seems such formulas are usually only isomorphisms for a restricted class of "dualizable" objects.
