The question says it all, but let me recall the definitions.
- A magma $(X, \cdot)$ is a set $X$ with a binary operation $\cdot \colon X \times X \to X$ (without any further assumptions like associativity).
- A morphism between two magmas $X$ and $Y$ is a map $f \colon X \to Y$ with $f(a \cdot b) = f(a) \cdot f(b)$ for all $a, b\in X$.
- A morphism $f$ is epi, respectively mono, if one can cancel it from the right, respectively left: $g \circ f = h \circ f \implies g=h$, respectively $f \circ g = f \circ h \implies g = h$.
Using free magmas with one generator shows that monomorphisms are injective. The question is whether epimorphisms are always surjective (in many other categories this fails, e.g., in the category of monoids).