Monomorphism and Epimorphism

Which of the following statements is/are true?

Ⅰ. A group homomorphism $f:G\to H$ is injective if and only if it is left-cancellative: given any group homomorphisms $g,h:F\to G,$ whenever $f\circ g=f\circ h,$ then $g=h.$

Ⅱ. A group homomorphism $f:G\to H$ is surjective if and only if it is right-cancellative: given any group homomorphisms $g,h:H\to K,$ whenever $g\circ f=h\circ f,$ then $g=h.$