$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 8.1.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\sigma _0: \Lambda ^{2}_{1} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets, corresponding to a diagram
8.7
\begin{equation} \begin{gathered}\label{equation:thin-filler-in-Corr} \xymatrix@C =50pt@R=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dl]^{u} \ar [dr]_{v} & & X_{2,2} \ar [dl] \\ & X_{0,1} & & X_{1,2} & } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{C}}$. Then any commutative diagram
8.8
\begin{equation} \begin{gathered}\label{equation:thin-filler-in-Corr2} \xymatrix@R =50pt@C=50pt{ X_{1,1} \ar [r]^-{u} \ar [d]^{v} & X_{1,2} \ar [d] \\ X_{0,1} \ar [r] & X_{0,2} } \end{gathered} \end{equation}
can be obtained from an extension of $\sigma _0$ to a $2$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. In particular, if $X_{1,2}$ and $X_{0,1}$ admit a pushout along $X_{1,1}$, then $\sigma _0$ can be extended to a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
Proof.
Let us identify $\operatorname{Tw}( \Delta ^2 )$ with the simplicial set $\operatorname{N}_{\bullet }(\overline{Q})$, where $\overline{Q}$ denotes the partially ordered set $\{ (i,j) \in [2]^{\operatorname{op}} \times [2]: i \leq j \} $. Under this identification, $\operatorname{Tw}( \Lambda ^{2}_{1} )$ corresponds to the simplicial subset $\operatorname{N}_{\bullet }(Q) \subseteq \operatorname{N}_{\bullet }( \overline{Q} )$, where $Q = \overline{Q} \setminus \{ (0,2) \} $, so $\sigma _0$ determines a diagram $\tau : \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{\mathcal{C}}$.
Set $Q_0 = Q \setminus \{ (0,0), (2,2) \} $ and $\tau _0 = \tau |_{ \operatorname{N}_{\bullet }( Q_0 ) }$. Lemma 8.1.4.4 is equivalent to the assertion that the restriction map $\operatorname{\mathcal{C}}_{ \tau / } \rightarrow \operatorname{\mathcal{C}}_{ \tau _0 / }$ is surjective on vertices. To prove this, it will suffice to show that the inclusion map $\operatorname{N}_{\bullet }(Q_0) \hookrightarrow \operatorname{N}_{\bullet }(Q)$ is right anodyne (Corollary 4.3.6.13), or equivalently that it is right cofinal (Proposition 7.2.1.3). This is a special case of Corollary 7.2.3.7, since the inclusion map $Q_0 \hookrightarrow Q$ has a left adjoint (given by $(0,0) \mapsto (0,1)$ and $(2,2) \mapsto (1,2)$).
$\square$