Magma category
In mathematics, the category of magmas (see category, magma for definitions), denoted by Mag, has as objects sets with a binary operations, and morphisms given by homomorphisms of operations (in the universal algebra sense).The category Mag has direct productss, so the concept of a magma object (internal binary operation) makes sense. (As in any category with direct products).
There is an inclusion functor from Set to Med to (inclusion) Mag as trivial magmass, with operationss: right, say, projections (bad references, we need projection maps) : x T y = y.
A very important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism.