  In this paper, we investigate a connection between convolution products for
quiver Hecke algebras and tensor products for quantum groups. We give a
categorification of the natural projection $ \pi_{\lambda, \mu} :
V_{\mathbb{A}}(\lambda)^\vee \otimes_{\mathbb{A}} V_\mathbb{A}(\mu)^\vee
\twoheadrightarrow V_\mathbb{A}(\lambda+ \mu)^\vee $ sending the tensor product
of the highest weight vectors to the highest weight vector in terms of
convolution products. When the quiver Hecke algebra is symmetric and the base
field is of characteristic $0$, we obtain a positivity condition on some
coefficients associated with the projection $\pi_{\lambda, \mu}$ and the upper
global basis, and prove several results related to the crystal bases. We then
apply our results to finite type $A$ using the homogeneous simple modules
$\mathcal{S}^T$ indexed by one-column tableaux $T$.
