An isomorphism in a category is a morphism $f \colon X \to Y$ for which there exists a morphism $g \colon Y \to X$ so that $gf = 1_X$ and $fg = 1_Y$. The objects $X$ and $Y$ are isomorphic whenever there exists an isomorphism between $X$ and $Y$, in which case one writes $X \cong Y$.