I'm using notation close to Street-Walters *"Yoneda structures"*.  
  

For any locally small category $\textbf{A}$ 
there are, of course, $\hat{\textbf{A}}:=\textbf{set}^{\textbf{A}^{op}}$ and 
$\check{\textbf{A}}:=(\textbf{set}^{\textbf{A}})^{op}$
as well as the corresponding Yoneda embeddings 
$Y(\textbf{A}):\textbf{A}\rightarrow\hat{\textbf{A}}$ and
$Z(\textbf{A}):\textbf{A}\rightarrow\check{\textbf{A}}$.  
  

For any locally small functor $F:\textbf{A}\rightarrow\textbf{B}$, there is a covariant $F$-*weighted-hom functor* $\textbf{B}(F,1):\textbf{B}\rightarrow\hat{\textbf{A}}$ as well as a contravariant one $\textbf{B}\langle1,F\rangle:\textbf{B}\rightarrow\check{\textbf{A}}$
and evaluation of $F$ on arrows can be encoded via a natural transformation from the covariant Yoneda 
$\chi^F:Y(\textbf{A})\Rightarrow\textbf{B}(F,1)F$ or via one from the contravariant one $\psi^F:Z(\textbf{A})\Rightarrow\textbf{B}\langle 1,F \rangle F$.  
  

It is a fact that, in $\textbf{Cat}$, the covariant $F$-weighted-hom is a left Kan extension along $F$ of the covariant Yoneda of its domain (SW Axiom 1 for the Yoneda structure of $\textbf{Cat}$) and that $F$ is the *absolute* left lifting of Yoneda along the $F$-weighted-hom (SW Axiom 2), namely:   

$(1) \hspace{12pt} (\textbf{B}(F,1),\chi^F)=lan\_FY(\textbf{A})$  

$(2) \hspace{12pt} (F,\chi^F)=LIF\_{\textbf{B}(F,1)} Y(\textbf{A})$  

By pedantically adapting the proof of $(1)$ I can see its contravariant version:   

$(1\*) \hspace{8pt}(\textbf{B} \langle 1,F \rangle,\psi^F)=lan\_FZ(\textbf{A})$  



Is $(1\*)$ a consequence of $(1)$ in the sense that there is a direct way, that I guess should pass through the underlying profunctors, to prove $(1\*)$ assuming $(1)$ ?   
  

Is there a contravariant version of $(2)$ ?



The contravariant side of the Yoneda stucture of Cat
I realize that the question was not precise, but I now hope to understand.   

For any locally small functor $F:\textbf{A}\rightarrow\textbf{B}$, evaluation of $F$ on arrows can be encoded via $\chi^F:Y(\textbf{A})\Rightarrow\textbf{B}(F,1)F$ or $\psi^F:\textbf{B}\langle 1,F \rangle F \Rightarrow Z(\textbf{A})$.  

SW Axioms 1 and 2 are   

$(1) \hspace{12pt} (\textbf{B}(F,1),\chi^F)=lan\_FY(\textbf{A})$  

$(2) \hspace{12pt} (F,\chi^F)=LIF\_{\textbf{B}(F,1)} Y(\textbf{A})$  

The same axioms applied to $F^{op}$ directly yield   

$(1\*) \hspace{8pt}(\textbf{B} \langle 1,F \rangle,\psi^F)=ran\_FZ(\textbf{A})$  

$(2\*) \hspace{12pt} (F,\psi^F)=RIF\_{\textbf{B}\langle 1,F \rangle} Z(\textbf{A})$  

(where $RIF$ means *absolute right lifting*). So the contravariant side of a Yoneda structure can be exploited in a 2-category with a duality involution. 


