Show that a category of algebras (and their homomorphisms) has the epimorphism surjectivity property iff no algebra has a proper epic subalgebra where the epimorphism surjectivity property (ES) means surjective homomorphisms are equivalent to epis and a subalgebra B $\leq$ A is epic whenever two homomorphisms $f,g:$A$ \to $C coincide on B implies they are equal.
Edit: The forward direction has been handled in the comments (by contrapositive). For the reverse direction, I was thinking of using a similar idea.
For the forward direction by contrapositive, suppose $\exists$ B < A s.t. B is epic. Consider the inclusion morphism $i:$B $\to$ A. Then $i$ is epi but cannot be onto since B is proper. Thus epi $\nRightarrow$ onto and so the category does not have ES.
Conversely, suppose epi $\nRightarrow$ onto. Let B $\leq$ A, $i:$B $\to$ A be the inclusion map which is epi but not onto, and $fi=gi$. Then $f=g$ so B is epic but since $i$ isn't onto we must have B < A. Hence there exists a proper epic subalgebra.