  The relation between the bosonic higher spin ${\cal W}_\infty[\lambda]$
algebra, the affine Yangian of $\mathfrak{gl}_{1}$, and the SH$^c$ algebra is
established in detail. For generic $\lambda$ we find explicit expressions for
the low-lying ${\cal W}_\infty[\lambda]$ modes in terms of the affine Yangian
generators, and deduce from this the precise identification between $\lambda$
and the parameters of the affine Yangian. Furthermore, for the free field cases
corresponding to $\lambda=0$ and $\lambda=1$ we give closed-form expressions
for the affine Yangian generators in terms of the free fields. Interestingly,
the relation between the ${\cal W}_\infty$ modes and those of the affine
Yangian is a non-local one, in general. We also establish the explicit
dictionary between the affine Yangian and the SH$^c$ generators. Given that
Yangian algebras are the hallmark of integrability, these identifications
should pave the way towards uncovering the relation between the integrable and
the higher spin symmetries.
