  Fix a finite-dimensional simple Lie algebra $\mathfrak{g}$ and let
$\mathfrak{g}_J\subseteq\mathfrak{g}$ be a Lie subalgebra coming from a Dynkin
diagram inclusion. Then, the corresponding restriction functor is not
essentially surjective on finite-dimensional simple $\mathfrak{g}_J$-modules.
In this article, we study Finkelberg-Tsymbaliuk's shifted quantum affine
algebras $U_q^{\mu}(\mathfrak{g})$ and the associated categories
$\mathcal{O}^{\mu}$ (defined by Hernandez). In particular, we introduce natural
subalgebras $U_q^{\nu}(\mathfrak{g}_J)\,{\subseteq}\,U_q^{\mu}(\mathfrak{g})$
and obtain a functor $\mathcal{R}_J$ from
$\mathcal{O}^{sh}\,{=}\bigoplus_{\mu}\mathcal{O}^{\mu}$ to
$\bigoplus_{\nu}(U_q^{\nu}(\mathfrak{g}_J)\text{-Mod})$ using the canonical
restriction functors. We then establish that $\mathcal{R}_J$ is essentially
surjective on finite-dimensional simple objects by constructing notable
preimages that we call inflations. We conjecture that all simple objects in
$\mathcal{O}^{sh}_J$ (which is the analog of $\mathcal{O}^{sh}$ for the
subalgebras $U_q^{\nu}(\mathfrak{g}_J)$) admit some inflation and prove this
for $\mathfrak{g}$ of type A-B or $\mathfrak{g}_J$ a direct sum of copies of
$\mathfrak{sl}_2$ and $\mathfrak{sl}_3$. We finally apply our results to deduce
certain $R$-matrices and examples of cluster structures over Grothendieck
rings.
