Fix the algebraic structures and the homomorphisms under consideration.
An Isomorphism \(\phi :A\to B\) is an iso:
a homomorphism admitting an
inverse homomorphism \(\phi ^{-1}:B\to A\), so that
\(\phi ^{-1}\circ \phi =\operatorname {id}_A\) and
\(\phi \circ \phi ^{-1}=\operatorname {id}_B\). We write \(A\cong B\) when such
an isomorphism exists.
This is the invertible-morphism definition of
[stacks2017stacks, Definition 4.2.4 and the following paragraph, Tag 0013].
An Endomorphism of \(A\) is a homomorphism \(\phi :A\to A\).
The set of all such homomorphisms is denoted \(\operatorname {End}(A)\).
Thus an individual endomorphism satisfies \(\phi \in \operatorname {End}(A)\).
An Automorphism of \(A\) is an endomorphism that is an
isomorphism. The set of all automorphisms is denoted \(\operatorname {Aut}(A)\).
These notations always refer to the chosen kind of homomorphism; for modules
over \(R\), the notation \(\operatorname {End}_R(M)\) specifies \(R\)-linear maps.
