## ยง Mnemonic for hom-tensor and left-right adjoints

- Remember the phrase
`tensor-hom`

adjunction, thus tensor is left adjoint. - Remember that the type of an adjunction is
`(f x -> y) -> (x -> g y)`

and here, `f`

is left adjoint, `g`

is right adjoint. Then see that currying is `((p, x) -> y) ->(x -> (p -> y))`

. Thus tensor is left adjoint, hom is right adjoint. - Remember that RAPL (right adjoints preserve limits); Then recall that tensoring a direct limit (a colimit) preserves the tensor, as a colimit retains torsion (example: prufer group has torsion, its components also have torsion; tensor can detect this by tensoring with $\mathbb Q$). On the other hand, tensoring of an inverse-limit (a limit) is not preserved: think of p-adics. Each of the components have torsion, but the p-adics do not. Thus, tensor DOES NOT preserve limits (inverse limits). And so, tensor CANNOT be right adjoint; tensor must be left adjoint.
- Since tensor is right exact, because it kills stuff, and could this destroy injectivity, it is left adjoint.