Two algebra structures on endomorphisms

Question

Let $(\mathcal{M}, \otimes, \mathbb{k})$ be a symmetric closed monoidal category, which in my application is the category of $dg$-modules over some commutative ring. Let $A$ be a bialgebra/bimonoid in $\mathcal{M}$ whose underlying object is dualizable. Then $A^{\vee}$ is also a bialgebra, and we have an isomorphism $A^\vee \otimes A \cong \mathcal{M}(A, A)$ (the right hand side being the internal hom). The left hand side is canonically an algebra since $A$ is an algebra and $A^\vee$ acquires an an algebra structures from the coalgebra structure on $A$. The right hand side is also an algebra, from the internalized composition operation of the category. How does this isomorphism interact with the two algebra structures?