  Given a unital $\boldsymbol{C}^{*}$-algebra $\mathcal{A}$, we prove the
existence of the coproduct of two faithful operator $\mathcal{A}$-systems. We
show that we can either consider it as a subsystem of an amalgamated free
product of $\boldsymbol{C}^{*}$-algebras, or as a quotient by an operator
system kernel. We introduce a universal $\boldsymbol{C}^{*}$-algebra for
operator $\mathcal{A}$-systems and prove that in the case of the coproduct of
two operator $\mathcal{A}$-systems, it is isomorphic to the amalgamated over
$\mathcal{A}$, free product of their respective universal
$\boldsymbol{C}^{*}$-algebras. Also, under the assumptions of hyperrigidity for
operator systems, we can identify the $\boldsymbol{C}^{*}$-envelope of the
coproduct with the amalgamated free product of the
$\boldsymbol{C}^{*}$-envelopes. We consider graph operator systems as examples
of operator $\mathcal{A}$-systems and prove that there exist graph operator
systems whose coproduct is not a graph operator system, it is however a dual
operator $\mathcal{A}$-system. More generally, the coproduct of dual operator
$\mathcal{A}$-systems is always a dual operator $\mathcal{A}$-system. We show
that the coproducts behave well with respect to inductive limits of operator
systems.
