For an \(\infty\)-operad \(\mathcal O\) and an operad map \([1] \otimes \mathbb E_0 \rightarrow\mathcal O\) corepresenting a morphism \(f \colon A \rightarrow B \in \mathrm{Alg}_{\mathbb E_0}(\mathcal O)\), we write \[\mathbb{T}^{\mathcal O}_2(f) \coloneqq \mathrm{Hom}_{\mathrm{Op}_{[1] \otimes \mathbb E_0/}}(\mathbb{T}_2, \mathcal O) \simeq \mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal O)}(A,A;B) \times_{\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal O)}(A;B)^2} \{(f,f)\}.\] for the space of \(\mathbb T_2\)-structures on \(f\).
For an \(\infty\)-operad \(\mathcal O\) and an operad map \(\mathbb E_0 \rightarrow\mathcal O\) corepresenting an \(\mathbb E_0\)-algebra \(A\) in \(\mathcal O\), we write \[\mathbb A^{\mathcal O}_2(A) \coloneqq \mathrm{Hom}_{\mathrm{Op}_{\mathbb E_0/}}(\mathbb A_2, \mathcal O) \simeq \mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal O)}(A,A;A) \times_{\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal O)}(A;A)^2} \{(\mathrm{id}_A,\mathrm{id}_A)\}.\] for the space of \(\mathbb A_2\)-structures on \(A\).
If clear from context, we will often drop the superscript \(\mathcal O\) indicating the ambient \(\infty\)-operad and simply write \(\mathbb T_2(f)\) and \(\mathbb A_2(A)\).