A morphism is a mathematical function that maps elements of one set, structure or category to another. In category theory, morphisms are also called arrows. For example, in the category of sets, a morphism is a function that maps elements of one set to another. In the category of groups, a morphism is a homomorphism that maps one group to another.
A functor is a structure-preserving mapping between categories. A functor maps objects and morphisms from one category to another in such a way that it preserves the properties of the categories. In other words, a functor maps objects in a way that respects the structure of the categories, and it maps morphisms in a way that preserves the composition and identity laws.
Category A: Category B:
a1 b1
/ \ / \
a2 a3 b2 b1
Morphisms in A: Morphisms in B:
f1: a1 -> a2 g1: b1 -> b2
f2: a1 -> a3 g2: b2 -> b1
f3: a2 -> a3
Functor F:
F(a1) = b1
F(a2) = b2
F(a3) = b1
F(f1) = g1
F(f2) = g2
F(f3) = g1