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: $$\mathcal{F}:\mathcal{Grpd}\hookrightarrow\mathcal{Cat}.$$ According to wikipedia, 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 "(co)limit" expression 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?
from Hot Weekly Questions - Mathematics Stack Exchange
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