The space \(\mathbb T_2(f)\) of \(\mathbb T_2\)-structures on a given \(\mathbb E_0\)-morphism \(f \colon A \rightarrow B\) in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal O)\) is compatible with composition and functorial: Given an operad map \(F \colon \mathcal O\rightarrow\mathcal P\), applying \(F\) induces a map of spaces \[\mathbb T_2(f) \xrightarrow{F(-)} \mathbb T_2(F(f)).\]
Similarly, any \(\mathbb E_0\)-morphism \(g \colon B \rightarrow C\) in \(\mathrm{Alg}_{\mathbb E_0}(\mathcal O)\) induces evident maps of spaces \[ \mathbb T_2(f) \xrightarrow{g \circ -} \mathbb T_2(g \circ f) \hspace{1cm} \mathbb T_2(g) \xrightarrow{- \circ (f,f)} \mathbb T_2(g \circ f) .\]
It will be useful to express this operation in terms of \(\infty\)-operads. Let \([2]\coloneqq \{0 <1<2\}\) denote the \(\infty\)-category (and the free \(\infty\)-operad on that \(\infty\)-category) corepresenting a pair of composable morphisms. Consider the following pushouts of \(\infty\)-operads 
corepresenting a pair of \(\mathbb E_0\)-morphisms \(A \xrightarrow{f} B \xrightarrow{g} C\) with a \(\mathbb T_2\)-structure on \(f\), \(g\) or \(g\circ f\), respectively. By Yoneda, the maps of spaces constructed above induce operad maps \[\mathbb T_2\sqcup_{\{1< 2\}} [2] \leftarrow \mathbb T_2\sqcup_{\{0<2\}}[2] \rightarrow\mathbb T_2 \sqcup_{\{0<1\}}[2].\]