The inclusion of groupoids into categories is left adjoint to the maximal-subgroupoid functor

Every functor $T:I\mathcal{G}\to \mathcal{C}$ sends inverses to inverses, so [F1] and [F2] force every image morphism into $\mathrm{Core}(\mathcal{C})$. Keeping the same object and morphism functions gives a unique factor $\bar{T}:\mathcal{G}\to \mathrm{Core}(\mathcal{C})$.

If $K:\mathcal{C}\to \mathcal{D}$ is a functor, it sends isomorphisms to isomorphisms and therefore restricts to $\mathrm{Core}(K):\mathrm{Core}(\mathcal{C})\to \mathrm{Core}(\mathcal{D})$. Identities and composites restrict unchanged, so $\mathrm{Core}$ is a functor.

The factorization in step 1.1 and inclusion give inverse bijections $\mathbf{C}\mathbf{a}\mathbf{t}(I\mathcal{G},\mathcal{C})\cong \mathbf{G}\mathbf{p}\mathbf{d}(\mathcal{G},\mathrm{Core}(\mathcal{C}))$. Their definitions by restriction show naturality in both variables.

The categories of small categories and small groupoids are locally small, so [L1] applied to step 2.1 gives $I⊣\mathrm{Core}$. The construction also covers the empty groupoid and empty category.