  Suppose $A\in \mathbb{R}^{m\times n}$ and consider the following canonical
systems of inequalities defined by $A$: $$ \begin{array}{l} Ax=b\\ x \ge 0
\end{array} \qquad \text{ and }\qquad A^T y - c \le 0. $$ We establish some
novel duality relationships between the Hoffman constants for the above
constraint systems of linear inequalities provided some suitable Slater
condition holds. The crux of our approach is a Hoffman duality inequality for
polyhedral systems of constraints. The latter in turn yields an interesting
duality identity between the Hoffman constants of the following box-constrained
systems of inequalities: $$ \begin{array}{l} Ax=b\\ \ell \le x \le u
\end{array}\qquad \text{ and }\qquad \ell \le A^T y - c \le u $$ for $\ell,
u\in \mathbb{R}^n$ with $\ell < u.$
