Eqiuvalence (Category Theory)

Let $\mathcal{C},\mathcal{D}:\mathbf{Cat}$. Say $\mathcal{C}\simeq \mathcal{D}$ are equivalent iff There is a functor $F:\mathcal{C}\to \mathcal{D}$ that is:

• Full Functor
• Faithful Functor
• Essentially Surjective Functor

Remarks

This is a strictly weaker notion of equivalence than isomorphism. There is a sense in which ‘real category theory’ is what is preserved by equivalence of categories.

Resources

https://math.stackexchange.com/questions/4476928/how-do-i-think-of-a-representable-functor