Morphism

Remarks

The morphism $g$ in clause 1 is unique: if $g'$ also works, then $g' = g' \circ (f \circ g) = (g' \circ f) \circ g = g$. It is called the inverse $f^{-1}$, and $A$ and $B$ are then isomorphic, written $A \cong B$. Example: in the category of groups, the morphisms are the group homomorphisms. The isomorphisms are the bijective homomorphisms, since the inverse map of a bijective homomorphism is again a homomorphism. Non-example: in the category of topological spaces with continuous maps, let $X$ be $[0, 1]$ with the discrete topology and $Y$ be $[0, 1]$ with the usual topology. The identity map $X \to Y$ is a continuous bijection, but its inverse $Y \to X$ is not continuous, so it is not an isomorphism. The endomorphisms of $A$ form a monoid $\End(A)$ under composition, and the automorphisms form a group $\Aut(A)$.