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The adjoint functor theorem as stated here and the special adjoint functor theorem (which can also both be found in Mac Lane) are both very handy for showing the existence of adjoint functors. First …

Suppose that $F\colon C\to D$ is a functor. Then there are many situations in which thinking of finding left and right adjoints to $F$ as solving approximation problems is very good intuition. So thes …

Provided that $X$ is quasi-compact and separated and $f$ is separated then what is true is that $Rf_\ast \colon \operatorname{D}(X) \to \operatorname{D}(Y)$ has a right adjoint $f^!$ where these are t …

At least if one takes labeled graphs (LGrphs) and labeled digraphs the functor you suggest, say D, is right adjoint to the forgetful functor which I'll call U. There is a canonical natural transformat …

There are still strictly speaking elements floating around in the following since we are using indexing sets but maybe it is better? Consider for a set $S$ and an abelian group $A$ the isomorphisms $$ …