ag.algebraic geometry - Is there a general projection formula for morphisms of ringed topoi?

In the context of sheaves of $mathcal O_X$-modules,
there is the following reference: Prop. 3.9.4 in Lipman's
Notes on derived functors and Grothendieck duality.
A closely related result is in Neeman's paper The Grothendieck duality theorem ...; see Prop. 5.3.

I'm not sure that analogous results should be expected to hold in arbitrary generality;
for example, both references place a restriction on the base scheme, and require quasi-coherence assumptions. (In some sense, one has to reduce to the locally free case,
where the statement is obvious. Quasi-coherent sheaves then admit locally free resolutions.
The proofs of the cited results apply some form of this argument in rather subtle and sophisticated ways.)