  In this paper we study properties of the secondary Hochschild homology of the
triple $(A,B,\varepsilon)$ with coefficients in $M$. We establish a type of
Morita equivalence between two triples and show that
$H_\bullet((A,B,\varepsilon);M)$ is invariant under this equivalence. We also
prove the existence of an exact sequence which connects the usual and the
secondary Hochschild homologies in low dimension, allowing one to perform easy
computations. The functoriality of $H_\bullet((A,B,\varepsilon);M)$ is also
discussed.
