right fibration is representable (cf. [@Lurie:2009un Proposition 4.4.4.5]). The Yoneda embedding[^7] ${\!\text{\usefont{U}{min}{m}{n}\symbol{'210}}\!}:{\mathcal}{C}\to {\mathscr{P}}({\mathcal}{C})$ identifies ${\mathcal}{C}$ with the full subcategory of ${\mathscr{P}}({\mathcal}{C})$ spanned by the representable presheaves (cf [@Lurie:2009un Proposition 5.1.3.1]); whence it follows that $\lambda'$ factors through ${\mathcal}{C}$, $$\begin{tikzpicture}
    {\matrix (m) [matrix of math nodes, row sep=3em,
  column sep=3em, text height=1.5ex, text depth=0.25ex]}{{\mathcal}{D}&&{\mathcal}{C}\\&{\mathscr{P}}({\mathcal}{C})&\\};
    \draw[->] (m-1-1) -- node[swap] {$\lambda'$} (m-2-2);
    \draw[right hook->] (m-1-3) -- node {${\!\text{\usefont{U}{min}{m}{n}\symbol{'210}}\!}$} (m-2-2);
    \draw[dashed,->] (m-1-1) -- node {$\lambda^R$} (m-1-3);
  \end{tikzpicture}$$ The Yoneda lemma implies that we have a weak equivalence of spaces $${\operatorname}{Hom}_{\mathcal}{C}\big(c,\lambda^R(d)\big)\cong \{c\}\times_{{\mathcal}{C}}{\mathcal}{M}\times_{{\mathcal}{D}^{op}}\{d\},$$ which depend naturally on $(c,d)\in {\mathcal}{C}^{op}\times{\mathcal}{D}$.

Similarly, an object of $m\in{\mathcal}{M}$ is called *left universal* if it is a terminal object of ${\mathcal}{M}\times_{{\mathcal}{C}}\{\lambda_C(m)\}$, and the right fibration is called a *left representable pairing*, if for each $c\in{\mathcal}{C}$, there exists a left universal object in the fibre ${\mathcal}{M}\times_{\mathcal}{C}\{c\}$ over $c$. As before, this determines a functor $\lambda^L:{\mathcal}{C}\to {\mathcal}{D}$; and the yoneda Lemma implies that we have weak equivalences of spaces $${\operatorname}{Hom}_{\mathcal}{C}\big(\lambda^L(c),d\big)\cong \{c\}\times_{{\mathcal}{C}}{\mathcal}{M}\times_{{\mathcal}{D}^{op}}\{d\}\cong{\operatorname}{Hom}_{\mathcal}{C}\big(c,\lambda^R(d)\big)$$ depending naturally on $(c,d)\in {\mathcal}{C}^{op}\times{\mathcal}{D}$. Indeed, $\lambda^R$ is a right adjoint to $\lambda^L$ (cf. [@Lurie:ZsNVU6Ru] or [@Lurie:2009un § 5.2.6] for more details).

Suppose that ${\mathcal}{M}\to {\mathcal}{C}\times{\mathcal}{D}^{op}$ and ${\mathcal}{M}'\to {\mathcal}{C}'\times{\mathcal}{D}'^{op}$ are two right representable right fibrations of $\infty$-categories, then a morphism of diagrams 