Category of sets has a strict initial object

Question

Since Set is cartesian closed, its initial object, $0 = \varnothing$ is strict. Consider the object $A = \{ 1, 2 \}$. There is no isomorphism (bijection) $f : A \to 0$, since the cardinalities are different. So how is it that $0$ is a strict initial object? There must be something I'm missing about isomorphisms.

Answer 1

There are no isomorphisms $A\to \varnothing$, this is true. However, this is not a problem as there aren't any morphisms $A\to \varnothing$ at all.

If an initial object $0$ in a category is not a strict initial object, there must be some morphism $f:X\to 0$ which is not an isomorphism. The only morphism in Set with $\varnothing$ as codomain is the identity map $\varnothing \to \varnothing$, which is an isomorphism. So $\varnothing$ is a strict initial object.