Hom, End, and Aut depend on the category

Core idea

For objects $A,B$ in a category $\mathcal C$ , write

\operatorname{Hom}_{\mathcal C}(A,B),\qquad \operatorname{End}_{\mathcal C}(A)=\operatorname{Hom}_{\mathcal C}(A,A),\qquad \operatorname{Aut}_{\mathcal C}(A)=\operatorname{End}_{\mathcal C}(A)^\times.

Here the last expression means the invertible elements of the endomorphism monoid under composition. The category is part of the question: an object's carrier alone does not specify its allowed maps.

What can change

For an algebra over a field extension $K/F$ , there are four natural views: a $K$ -vector space, an $F$ -vector space, a $K$ -algebra, and an $F$ -algebra. Forgetting multiplication permits more maps; restricting scalars can also permit more maps. See the comparison of endomorphism spaces.

For Jordan algebras, the unit convention changes Hom and End, even though all three categories have the same automorphisms of a unital object.

For a Lie group such as $SL(2,\mathbb C)$ , use smooth real group maps or holomorphic group maps. Its Lie algebra $\mathfrak{sl}_2(\mathbb C)$ has distinct real-linear and complex-linear endomorphisms. A matrix group is not generally a vector space.

What the catalogue records

A record marked complete describes the whole requested Hom-set, endomorphism monoid, or automorphism group under its stated conditions. A partial record gives a verified family, constraint, or example, without claiming exhaustiveness. A missing record says not catalogued. It does not say that no maps exist.

This distinction is visible in the finite-field examples: an empty field Hom-set can become a singleton containing the zero map after switching to arbitrary ring homomorphisms.

Source and target	Field homomorphisms	Arbitrary ring homomorphisms
$\mathbb F_2\to\mathbb F_4$	the unique prime-field inclusion	inclusion and zero
$\mathbb F_2\to\mathbb F_3$	none	zero only
$\mathbb F_4\to\mathbb F_8$	none	zero only
$\mathbb F_4\to\mathbb F_4$	identity and $x\mapsto x^2$	those two and zero