  We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into
a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common
generalization of two natural, but seemingly unrelated, embeddings, one on the
quantum affine Schur algebra level and the other on the non-quantum level. The
embedding on the quantum affine Schur algebras is used extensively in the
analysis of canonical bases of quantum affine $\mathfrak{sl}_n$ and
$\mathfrak{gl}_n$. The embedding on the non-quantum level is used crucially in
a work of Riche and Williamson on the study of modular representation theory of
general linear groups over a finite field. The same embedding is also used in a
work of Maksimau on the categorical representations of affine general linear
algebras. We further provide a more natural compatibility statement of the
embedding on the idempotent version with that on the quantum affine Schur
algebra level. A $\mathfrak{gl}_n$-variant of the embedding is also
established.
