Let $\mathscr{C}$ and $\mathscr{D}$ be closed symmetric monoidal categories. We fix a strong symmetric monoidal functor $f^*:\mathscr{D}\to\mathscr{C}$ with a right adjoint $f_*:\mathscr{C}\to\mathscr{D}$. In this context, there is a formal isomorphism $\hom(Y,f_*X)\cong f_*\hom(f^*Y,X)$.

Now, suppose that we're given a second adjoint pair $(f_!,f^!)$ relating $\mathscr{C}$ and $\mathscr{D}$. It would be nice if we had a internal adjunction $\hom(f_!X,Y)\cong f_*\hom(X,f^!Y)$.

**While this holds in most 6 functor formalisms, this doesn't follow formally from our suppositions.**

In *Isomorphisms between left and right adjoints*, H. Fausk, P. Hu, and J.P. May affirm that it suffices to give the existence of one of the three arrows
$$f_*\hom(X,f^!Y)\to \hom(f_!X,Y),\quad \hom(f^*Y,f^!Z)\to f^!\hom(Y,Z),\quad\pi:Y\otimes f_!X\to f_!(f^*Y\otimes X)$$
for all three to exist. Moreover, if one of them is an isomorphism, so are them all.

**I wonder what condition holds in practice for us to have at least the existence of these morphisms.**

Perhaps the fact that we usually have a morphism $f_!\to f_*$ suffices for us to construct these morphisms? (That's how we sometimes construct the projection formula. But we usually have that this morphism is injective and this doesn't hold for $D$-modules, for example.)

Perhaps a base change theorem suffices as in Ryan Reich's answer in Ubiquity of the push-pull formula? (This answer doesn't completely solve my problem since the formula $(g\times h)_!(X\boxtimes Y)\cong g_! X\boxtimes h_! Y$ is not clear to me as well.)

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