Category of associative algebras with arbitrary homomorphisms

Fix a field $k$ . The category of associative $k$ -algebras with arbitrary homomorphisms has $k$ -vector spaces with a bilinear associative product as objects. A morphism $f:A\to B$ is $k$ -linear and satisfies

f(xy)=f(x)f(y).

Objects need not have units, and units are not required to be preserved when present. Identities and composition are the corresponding linear maps.

Unit-preserving variant

The unital category requires units in the objects and their preservation by morphisms. In the arbitrary-map category, the zero map and a matrix-corner inclusion are allowed. Both categories have a forgetful functor to the category of vector spaces, but most linear maps fail to preserve the product.