I am trying to figure out what should the following definition correspond to in the setting of quantum affine algebra: $$ X \circ Y = Ind_{\beta, \gamma}^{\beta+\gamma} X \boxtimes Y \quad (1) $$ This is on page 13 of the paper Quiver Hecke Algebras and Categorification.
By the quantum affine Schur-Weyl duality and the results in Quantum Grothendieck rings and derived Hall algebras, the modules $X, Y$ correspond to some modules of quantum affine algebra $U_q(\widehat{\mathfrak{g}})$. Denote by $F(X)$ the corresponding $U_q(\widehat{\mathfrak{g}})$-module in $\mathcal{C}_{\beta}$, $F(Y)$ the corresponding $U_q(\widehat{\mathfrak{g}})$-module in $\mathcal{C}_{\gamma}$, where $\mathcal{C}_{\beta}$, $\mathcal{C}_{\gamma}$ are subcategories of the category of finite dimensional $U_q(\widehat{\mathfrak{g}})$-modules defined in Lemma 2.26 of AFFINE HIGHEST WEIGHT CATEGORIES AND QUANTUM AFFINE SCHUR-WEYL DUALITY OF DYNKIN QUIVER TYPES.
The left hand side $X \circ Y$ of the equation (1) corresponds to the tensor product in the setting of quantum affine algebra.
In the setting of quantum affine algebra, let us formally write the following equation corresponding to (1): $$ F(X) \otimes F(Y) = Ind_{\beta, \gamma}^{\beta+\gamma} F(X) \boxtimes F(Y). $$
How to understand $Ind_{\beta, \gamma}^{\beta+\gamma} F(X) \boxtimes F(Y)$ in the setting of quantum affine algebras? Thank you very much.
