TITLE: Grothendieck's yoga of six operations - in relatively basic terms?
QUESTION [34 upvotes]: I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper inverse/direct images functors $f_!\dashv f^!$, and derived tensor product/internal hom $(-\otimes) A\dashv \mathsf{hom}(A,-)$.
I thought the pairs $f^\ast\dashv f_\ast$ and the tensor-hom adjunction looked familiar from the category of sets, although in that case the inverse image functor is right adjoint to the direct image functor (see this question). After poking around on google I found out about "Grothendieck's six operations". In every place I looked though, these were discussed at a level far beyond mine and in contexts I do not know - quasi-coherent sheaves, spectra, étale cohomology...

What is the idea and significance behind Grothendieck's six operations in general, and in the context of sheaves (over topological spaces) in particular?

I don't mind basic concepts from enriched category theory if they help.

REPLY [5 votes]: I don't have the time to expand the answer in a detailed but succinct account of those, but if you're more on the category theory side of the story I can throw some references which contain some information: it should appear evident why they are kind of related to the question

J-B.  Calmès and J. Hornbostel, "Tensor-triangulated categories and dualities", Theory and Applications of Categories, Vol. 22, 2009, No. 6, pp 136-198.
Ryan Reich, "Obvious natural morphisms of sheaves are unique", Theory and Applications of Categories, Vol. 29, 2014, No. 4, pp 48-99.
F. Hörmann, "Fibered Multiderivators and (co)homological descent" that contains all the prerequisites for
F. Hörmann, "Six Functor Formalisms and Fibered Multiderivators"