$$\begin{tikzpicture}
    {\matrix (m) [matrix of math nodes, row sep=3em,
  column sep=3em, text height=1.5ex, text depth=0.25ex]}{{\mathcal}{M}&{\mathcal}{M}'\\{\mathcal}{C}\times{\mathcal}{D}^{op}&{\mathcal}{C}'\times{\mathcal}{D}'^{op}\\};
    \draw[->] (m-1-1) -- (m-2-1);
    \draw[->] (m-1-2) -- (m-2-2);
    \draw[dashed,->] (m-1-1) -- node {$\gamma$} (m-1-2);
    \draw[dashed,->] (m-2-1) -- node {$\alpha\times\beta$} (m-2-2);
  \end{tikzpicture}$$ is called *right representable* if it takes right universal objects to right universal objects.

The $\infty$-category of right-representable pairings ${\operatorname}{CPair}^R\subseteq{\operatorname}{CPair}$ is defined to be the subcategory whose objects are equivalent to right representable pairings, and whose morphisms are equivalent to right representable morphisms. The $\infty$-category of left-representable pairings ${\operatorname}{CPair}^L\subseteq{\operatorname}{CPair}$ is defined analogously.

\[lem:CPairClosed\] Let ${\operatorname}{CPair}^R\subseteq{\operatorname}{CPair}\subseteq {\operatorname}{Fun}(\Lambda_0^2,{{\mathscr{C}\mathrm{at}}}_\infty)$ be the $\infty$-categories defined in [@Lurie:ZsNVU6Ru]. Then both subcategories are closed under small limits.

Since ${\operatorname}{CPair}\subseteq {\operatorname}{Fun}(\Lambda_0^2,{{\mathscr{C}\mathrm{at}}}_\infty)$ is a reflective localization (cf. [@Lurie:ZsNVU6Ru Remark 4.2.9]), it is closed under small limits; so we need only show that ${\operatorname}{CPair}^R\subseteq{\operatorname}{CPair}$ is also closed under small limits.

Let $p:K^{op}\to {\operatorname}{CPair}^R$ be a diagram (for which we wish to compute the limit). The composite functor $K^{op}\to {\operatorname}{CPair}^R\to {\operatorname}{CPair}$ is classified by a diagram of simplicial sets $$\begin{tikzpicture}
{\matrix (m) [matrix of math nodes, row sep=3em,
  column sep=3em, text height=1.5ex, text depth=0.25ex]}{{\mathcal}{M}_p   & {\mathcal}{D}_p^{op}\\ {\mathcal}{C}_p & K^{op}\\};
\draw[->] (m-1-1) -- node {$\lambda_C$} (m-1-2);
\draw[->] (m-1-1) -- node [swap] {$\lambda_D$} (m-2-1);
\draw[->] (m-1-2) -- 