The unit-counit, hom-set, unit-universal, and counit-universal encodings of an adjunction are equivalent

Descriptions 1 and 2 determine one another by [L2], including the recovery formulas obtained by transposing identity morphisms.

Description 1 gives descriptions 3 and 4 by [L3].

Conversely, from description 3, put $u^{\mathrm{♭}}:=G(u)\circ {\eta }_c$ for $u:Fc\to d$; the unique-factorisation property of $(Fc,{\eta }_c)$ says exactly that this is a bijection onto the morphisms $c\to Gd$, with inverse the unique factorisation. Define ${\epsilon }_d:FGd\to d$ as the unique morphism with $G({\epsilon }_d)\circ {\eta }_{Gd}=1_{Gd}$, which is the second triangle identity. For $g:d\to d^{\mathrm{\prime }}$ the two morphisms $g\circ {\epsilon }_d$ and ${\epsilon }_{d^{\mathrm{\prime }}}\circ FG(g)$ have the same factorisation datum, since $G(g)G({\epsilon }_d){\eta }_{Gd}=G(g)$ and $G({\epsilon }_{d^{\mathrm{\prime }}})GFG(g){\eta }_{Gd}=G({\epsilon }_{d^{\mathrm{\prime }}}){\eta }_{G{d}^{\mathrm{\prime }}}G(g)=G(g)$; uniqueness makes $\epsilon$ natural. Likewise $G({\epsilon }_{Fc})GF({\eta }_c){\eta }_c=G({\epsilon }_{Fc}){\eta }_{GFc}{\eta }_c={\eta }_c=G(1_{Fc}){\eta }_c$, so uniqueness gives the first triangle identity ${\epsilon }_{Fc}F({\eta }_c)=1_{Fc}$. This recovers a unit and counit satisfying [L1].

The dual construction from description 4 defines ${\eta }_c:c\to GFc$ as the unique morphism with ${\epsilon }_{Fc}\circ F({\eta }_c)=1_{Fc}$, proves its naturality from terminality of $(Gd,{\epsilon }_d)$ in $(F\downarrow d)$ by the argument of step 1.3 read in the opposite categories, and yields the second triangle identity the same way. It therefore gives the same unit-counit data.

Steps 1.1 through 2.1 establish all implications. Since [L3] is expressed by unique individual factorisations, descriptions 1, 3, and 4 remain meaningful without local smallness, whereas [L2] explicitly uses hom-sets.