A left adjoint exists exactly when chosen initial objects are supplied in every comma category

If a left adjoint $F$ is supplied, [L1] gives the required chosen initial object $(Fc,{\eta }_c)$ for every $c$.

Conversely, suppose such initial objects are supplied. For $a:c\to c^{\mathrm{\prime }}$, initiality gives a unique $F(a):Fc\to F{c}^{\mathrm{\prime }}$ satisfying $G(F(a))\circ {\eta }_c={\eta }_{c^{\mathrm{\prime }}}\circ a$.

The identity $1_{Fc}$ satisfies the defining equation for $F(1_c)$, so uniqueness gives $F(1_c)=1_{Fc}$.

If $a:c\to c^{\mathrm{\prime }}$ and $b:c^{\mathrm{\prime }}\to c^{\mathrm{\prime }\mathrm{\prime }}$, then $F(b)F(a)$ satisfies the defining equation for $F(ba)$; uniqueness gives $F(ba)=F(b)F(a)$. Thus $F$ is a functor and $\eta$ is natural.

For every $f:c\to Gd$, initiality supplies a unique $\widehat{f}:Fc\to d$ with $G(\widehat{f}){\eta }_c=f$. Steps 2.1 and 2.2 make $F$ a functor and $\eta$ natural, so $(Fc,{\eta }_c{)}_c$ is a natural family of universal arrows from each $c$ to $G$; by [L2] that data is an adjunction, so $F⊣G$.

Any functor action compatible with the chosen initial objects must satisfy the equation in step 1.2 and is therefore equal to this one. Passing to opposite categories proves the terminal-object criterion for right adjoints.