Adjunctions are everywhere, and it is (sometimes) somewhat easy to remember/divine when a pair of functors is an adjunct pair. What is harder, however, for me at least, is remembering which functor is the left adjoint and which the right. Is there some good way of working out which adjoint is which?
I know that we can always see if a functor commutes with small limits or etc., but this can be a tiresome calculation at times. I was hoping for something similar to 'colimits are gluings and limits are magnifications' in that it isn't at all formal or always true, and misses a lot of detail, but it helps remember instantly e.g. whether intersections are given by a product or coproduct in the category of open subsets of some space, and similarly for unions.
For example, we have $\mathrm{Free}\dashv\mathrm{Forget}$, $|\cdot|\dashv\mathrm{Sing}$, and $\otimes\dashv\mathrm{Hom}$. How can we 'remember' which side of the adjunction is which in these pairs?
I don't know if this helps much, but I remember which one is which by the formula $$\hom_C(L(-),-)\cong \hom_D(-,R(-))$$ where $L$ is the left adjoint and $R$ is the right adjoint. Given a functor $F$ that is part of an adjunction, it is often easier (at least for me) to figure out to which side of the above isomorphism $F$ belongs, i.e. in which direction the morphisms are supposed to go. Of course in order for this to help you have to know something about the functors in question.