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I asked this same question on MathUnderflow two weeks ago but didn't receive any answer. Now that I am thinking more, it feels like the most suitable place for this question is here.

I have a good grasp of all different definitions/interpretations of adjoint functors, but still do not know have to interpret the left or right adjoint of a give functor, when it exist. It would be a easy to explain my question through an example.

For a particular example consider the inclusion of groupoids into small categories: $$\mathcal{F}:\mathcal{Grpd}\hookrightarrow\mathcal{Cat}.$$ This functor has left adjoint $\mathcal{L}$ which freely invert all existing morphisms of a category. Also it has a right adjoint $\mathcal{R}$ which extract the (maximal) subcategory of all isomorphisms, called "core groupoid" of a category. In some scene this bi-adjunction align with the free-forgetful philosophy. Also we can compute these adjoint functors explicitly as pointwise Kan extensions, then $$\mathcal{L}(\mathcal{C})=\text{Ran}_{\mathcal{F}}(\text{id}_{\mathcal{Grpd}})(\mathcal{C})=\lim(\mathcal{C}\downarrow\mathcal{F}\xrightarrow{\Pi_{\mathcal{C}}}\mathcal{Grpd}\xrightarrow{\mathcal{F}}\mathcal{Cat}).$$ But I don't understand how to interpret this "limit" as a localization process and same for the right adjoint $\mathcal{R}.$ Can we derive it from this formula? If not, how can we see it?

In general, If we know some adjoint of a given functor exist, what is the process to understand it's effect/outcome?

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    $\begingroup$ If I have a functor $G: A\to B$, a left adjoint $F$ is a functor $B\to A$ that, for each $b\in B$ picks out the best approximation of $b$ by $G$(someone), via the unit $b\to GFb$. "best approximation" here is in the sense of "mapping out of $b$, no one will do a better job" (that's why left adjoints are often given by colimits, and right adjoints, where the story is dual, by limits). You already know that probably (hence a comment, not an answer), but that's one of the usual ways to think about adjoints $\endgroup$
    – Maxime Ramzi
    Jul 9, 2020 at 8:18

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Nice question Bumblebee. So, let us start with some "metaphysics of adjointness":

THE LEFT AND RIGHT ADJOINTS TO A FUNCTOR

$ \mathcal{F}:\mathcal{C}\hookrightarrow\mathcal{D}$

ARE THE FREE (LEFT) AND CO-FREE (RIGHT) WAYS TO GO BACK FROM $D$ TO $C$.

If you choose some easy examples, for instance $C=Top$ and $D=Set$ and the functor is simply the forgetful functor which "forgets" the topological structure, Left and Right start from a given set and endow it with a topology, in the most economic way (trivial topology) or in the most rigid one (discrete) . Same happens if you replace $Top$ with $Groups$ (or any other algebraic category).

Now, not all functors which have adjoints are forgetful functors, so matters are slightly more subtle sometimes, but the "general metaphysics" of adjointness still holds true.

Now the second part of your question, the scary formula for your example: rather that filling this page with calculations, I want to give you the heuristics (so far I have told you what adjoints are, not whether they exist or how they are calculated).

Here, I use the second "metaphysical principle of adjointness", namely this:

THINK OF CATEGORIES AS GENERALIZED ORDERS, AND OF ADJOINTNESS AS GENERALIZED GALOIS CONNECTIONS.

If you look up Galois connections (see here), how they are defined and calculated, you will also understand cats, by generalizing. Same exact story.....

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    $\begingroup$ Thank you for this nice explanation. Adjoints were a huge dark spot in my knowledge. I will get back to you later, as needed for more details. $\endgroup$
    – Bumblebee
    Jul 11, 2020 at 17:20
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    $\begingroup$ Please do. But not before you have digested galois connections. If you follow that tip, you will "get it" also for the more general notion of adjointness $\endgroup$
    – Mirco A. Mannucci
    Jul 11, 2020 at 17:42
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If you start with a category and only consider what you can see by looking at functors from groupoids, well you’ll only see the invertible morphisms. So the right adjoint is the core.

If you start with a category and only consider what you see by looking at functors to groupoids, well that’s a little harder because a non-invertible morphism can be sent to an invertible one. So to get the left adjoint you have to try to add formal inverses.

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    $\begingroup$ Thank you for the answer. How do you think of left or right adjoint functors in general? $\endgroup$
    – Bumblebee
    Jul 9, 2020 at 3:12

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