Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a functor. The following conditions are equivalent:
$F$ is full and faithful and has a full and faithful left adjoint $G$.
$F$ has a left adjoint $G$ and the two canonical natural transformations $\eta:1_{\mathcal{D}}\Rightarrow F\circ G$ and $\varepsilon:G\circ F\Rightarrow 1_{\mathcal{C}}$ are isomorphisms.
There exists a functor $G:\mathcal{D}\rightarrow\mathcal{C}$ such that $1_{\mathcal{D}}\cong F\circ G$ and $1_{\mathcal{C}}\cong G\circ F$.
$F$ is full and faithful and for each $D\in\mathcal{D}$ there exists $C\in\mathcal{C}$ such that $D\cong F(C)$.
the dual condition of 1.
the dual condition of 2.
What are the duals of the statements 1 and 2? Are they the following?
1*. $F$ is full and faithful and has a full and faithful right adjoint $G$.
2*. $F$ has a right adjoint $G$ and the two canonical natural transformations $\eta:F\circ G\Rightarrow 1_{\mathcal{D}}$ and $\varepsilon:1_{\mathcal{C}}\Rightarrow G\circ F$ are isomorphisms.
A left adjoint for $F:C^{op}\to D^{op}$ is a functor $G:D^{op}\to C^{op}$ with a natural isomorphism $$ \hom_{C^{op}}(Ga,b) \cong \hom_{D^{op}}(a,Fb) \text{ for all } a\in D, b\in C. $$
Equivalently, this is a natural isomorphism $$ \hom_{C}(b,Ga) \cong \hom_{D}(Fb,a) \text{ for all } a\in D, b\in C. $$
This shows that in fact $G$ is a right adjoint for $F$, i.e. left and right adjoints are dual concepts. This explains why 1* and 2* are correct.